Formulae Handbook for CBSE Class 10 Science and Maths

Formulae Handbook for CBSE Class 10 Science and Maths Features of this eBook:     

Can revise entire Term 1 or 2 Maths / Science content in 20 Minutes Quickly revise all formulas Revise important points of Each Chapter of Class 10 Maths and Science Included important diagrams of CBSE Class 10 Science Included additional Algebraic formulas, Exponents, Areas and volume formulae for better revision

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Formulae Handbook for CBSE Class 10 Science and Maths

Table of Contents

TERM-1 Science 1. Chemical Reactions and Equations ................................................... .......................................... 3 2. Acids, Bases and Salts ...................................................... .......................................................................................................... ..........................................................10 ......10 3. Metals & Non-Metals ............................................. ..................................................... .................................................................... ............... 17 4. Life Processes ................................................ ...................................................... ............................................................................ ...................... 25 5. Control and Co-ordination Co-ordination ........................................................ ......................................................................................................... ................................................. 31 6. Electricity .................................................................................................................................. 35 7. Magnetic Effects of Electric Current ................................................. ........................................ 40 8. Sources of Energy ..................................................................................................................... 45 Maths 1. Real Number ..................................................................................... ........................................ 48 2. Polynomials................................................... ...................................................... ............................................................................ ...................... 49 3. Pair of Linear Equations in Two Variables ................................................ ............................... 52 4. Triangles ................................................................................................................................... 54 5. Introduction to Trigonometry Trigonometry.................................................. ................................................. 63 6. Statistics .................................................................................................................................... 67 TERM-2 Science 1. Carbon and its Compounds ....................................................... ........................................................................................................ ................................................. 71 2. Periodic Classification of Elements Elements .................................................................... ...................... 78 3. How Do Organisms Reproduce?.............................................. .................................................. 81 4. Heredity and Evolution.................................................... ........................................................................................................ ......................................................... ..... 85 5. Reflection and Refraction ................................................ .................................................... ......................................................... ..... 89 6. Human Eye and the Colourful World ............................................... ........................................ 96 7. Management of Natural Resources................................................... ...................................... 100 8. Our Environment Environment................................................... ..................................................... ................................................................. ............ 102 Maths 1. Quadratic Equations............................................... ..................................................... .................................................................. ............. 105 2. Arithmetic Progressions ........................................ ....................................................... .................................................................. ........... 107 NCERT Solutions

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3. Circles (Tangents to a Circle) .................................................... ................................................................................................... ............................................... 110 4. Constructions ................................................ ...................................................... ........................................................................... ..................... 113 5. Areas Related to Circles ........................................................................................................... 117 6. Surface Areas & Volumes ................................................. .................................................... ....................................................... ... 120 7. Applications of Trigonometry ............................................................................. ..................... 122 8. Co-ordinate Geometry Geometry ........................................... ..................................................... .................................................................. ............. 123 9. Probability..................................................... ........................................................................................................... ........................................................................... ..................... 126 126 MATHEMATICAL MATHEMATICAL FORMULAE.................................................. ................................................ 128

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TERM-1

Key Points and Concepts A complete chemical equation represents the reactants, products, conditions of reaction and their physical states symbolically.

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Oxidation: loss of electrons. Reduction: gain of electrons.

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Rust is mainly hydrated iron (III) oxide, Fe2O3 x . H2O.

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Reactant: The substance which takes part in a chemical reaction is called reactant. For example, in the breaking up of calcium carbonate into calcium oxide and carbon dioxide, calcium carbonate is the reactant. Similarly, sodium and water are the reactants when they react. Product: A product is a new substance formed in a chemical reaction. For example, hydrogen and sodium hydroxide are the products of the reaction between sodium and water. 2Na + 2H 2O  2NaOH + H 2 sodium

sodiumhy droxide

water

hydrogen

Similarly, in the breaking up of calcium carbonate, calcium oxide and carbon dioxide are the products. CaCO3 calciumcarbonate





CaO

calcium oxide

CO2 carbondioxide

Valency: The number of electrons shared by an atom is called its valency. It is also called the combining capacity of an atom, e.g., Cl atom can share one valence electron, its valency is 1, Oxygen can share two valence electrons, its valency is 2. Nitrogen can share 3 valence electrons, its valency is 3, Carbon can share 4 valency electrons, and therefore its valency is 4 and so on. Element C

Cl

Valency 4

1

i.e., CCl4 is formula

Some more examples are: Element H

O H

Cl H

S N

H P

Cl N

O

Valency 1

2

1

2

1

1

2

H2O

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1 HCl

1 H2S

3

NH3

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3

PCl3

5

N2O5

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Some Common Monoatomic Ions +1 Charge Name of ion Copper ion (Cuprous ion)

+2Charge Formula

Name of ion

+3 Charge Formula

Name of ion

Formula

Cu+

Barium ion Cobalt ion

Ba2+ Co2+

Aluminium ion Auric ion

Al3+ Au3+

Potassium ion

K +

Strontium ion

Sr2+

Chromium (III) ion

Cr3+

Silver ion

Ag+

Sodium ion Lithium ion

Na+ Li+

Aurous

Au+

– 1 Charge Name of ion Bromide ion Chloride ion Fluoride ion Iodide ion

Iron (II) ion (Ferrous ion) *Copper (II) ion *Lead (II) ion Cadmium ion Magnesium ion Manganese (II) ion

Br– Cl– F– I–

Cu 2+ Pb 2+ Cd2+ Mg2+

Iron (III) ion (Ferric ion) Scandium ion Arsenic ion Bismuth ion Antimony ion

Fe3+ Sc3+ As 3+ Bi3+ Sb3+

Mn2+

*Mercury (I) ion

Hg22

Zinc ion

Zn2+

– 2Charge Name of ion Oxide ion Sulphide ion

Formula

Fe2+

Formula O2– S2–

– 3 Charge Name of ion Nitride ion Phosphide ion Boride ion

Formula N3– P3– B3–

Note: These elements show more than one valency. So a Roman numeral shows their valency in a bracket. Some Common Polyatomic Ions – 1 Charge Name of ion Hydrogen carbonate or bicarbonate ion Hydrogen sulphate or (bisulphate ion)

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Formula

– 2Charge Name of ion

HCO3

Carbonate ion Manganate ion

MnO24

HSO 4

Thiosulphate ion

S2O3

Silicate ion

Formula CO3  2

2

2 SiO 3

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– 3 Charge Name of ion Phosphate ion Arsenate ion

Arsenite ion

Formula P O34 AsO34

AsO33

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Hydroxide ion

OH–

Sulphate ion

SO 24

Nitrate ion

 NO3

Sulphite ion

SO 32

Chlorate ion

ClO3

Chromate ion

Nitrite ion

 NO2

Permanganate ion

MnO4

Acetate ion Cyanide ion Hypophosphite ion Meta aluminate ion

CH3COO–

Phosphite ion

P O33

CrO24

Borate ion

BO33

Dichromate ion

Cr 2O27

Ferricyanide ion

Hydrogen phosphate ion

HPO4

Oxalate ion

C2O24

2



CN– H 2PO2

– 4 Charge

 AlO2

Carbide ion Ferrocyanide ion

+1 Charge Ammonium ion

[Fe(CN) 6]3–

C4– [Fe(CN) 6]4–

NH4

Rules for writing chemical equation: Certain rules have to be followed while writing a chemical equation. 1. The reactants taking part in the reaction are written in terms of their symbols or molecular formulae on the left-hand side of the equation. 2. A plus (+) sign is added between the formulae of the reactants. 3. The products of reaction are written in terms of their symbols or molecular formulae on the right-hand side of the equation. 4. A plus (+) sign is added between the formulae of the products. 5. In between the reactants and the products an arrow sign ( ) is inserted to show which way the reaction is occurring. A + B  C + D In this chemical equation, A and B are the reactants, and C and D are the products. The arrow indicates that the reaction proceeds towards the formation of C and D.

Steps in Balancing of Chemical Equations: A number of steps are involved in balancing a chemical equation, e.g., Na + H2O  NaOH + H2 Step-1: Examine the number of atoms of different elements present in unbalanced equations. Number of atoms in reactants

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Number of atoms in products

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Na

1

1

H

2

3

O

1

1

Step-2: Pick an element to balance the equation. In the above equation Na and O are balanced, Hydrogen is not. Step-3: To balance Hydrogen on both sides we need to multiply H 2O by 2 which makes Hydrogen atoms equal to 4 on the reactants' side. To make Hydrogen 4 on the products' side, multiply NaOH by 2. Now oxygen has become 2 on both side. But Sodium atoms has become two on the products' side. Multiply Na by 2 on the reactants side so that they become equal on both side. The steps are as follows: (i) Na + 2 H2O  NaOH + H2 (ii) Na + 2 H 2O  2NaOH + H2 (iii)2 Na + 2 H2O  2NaOH + H2 The equation is now balanced. Example: Fe + H2O

Fe3O4 + H2

Step-1: Element

Number of atoms in reactants

Number of atoms in products

Fe

1

3

H

2

2

O

1

4

Step-2: Pick up the compound which has the maximum number of atoms whether a reactant or a product, and in that compound select the elements which has the highest number of atoms, e.g., we select Fe 3O4 in the above equation: To balance oxygen atoms, In reactants

In products

Initial

1 (in H2O)

4 (in Fe3O4)

To balance

1×4

4×1

To equalise the number of atoms, we put the coefficient on the left side of the formula. A coefficient is a small whole number, like coefficients used in algebraic equations. You must keep in mind that we can put coefficients but we cannot change the subscripts in the formula, i.e., to balance Oxygen atoms, we can put the coefficient 4 as 4 H 2O and not H2O4 or (H2O)4. Now the partly balanced equation becomes as follows: Fe(s) + 4 H2O(g)  Fe3O4(s) + H2(g) (Partly balanced) Step-3: Pick up the second element to balance this partly balanced equation. Let us try to balance hydrogen atoms.

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In partly balanced equation. Atoms of Hydrogen. In reactants

In products

Initial

8 (in 8 H2O)

2 (in H4)

To balance

8×1

2×4

To equalise the number of Hydrogen atoms, we use 4 as the coefficient of H2 in the products. Fe(s) + 4 H2O(g)  Fe3O4(s) + 4 H2 Step-4: Pick up third element to be balanced. The element which is left to be balanced is Fe. In reactants

In products

Initial

1 (in Fe)

3 (in Fe3O4)

To balance

1×3

3×1

To equalise, we use 3 as coefficient of Fe in reactants. 3Fe + 4H2O  Fe3O4 + 4H2 Atoms

In reactants

In products

Fe

3

3

H

8

8

O

4

4

The equation is balanced because atoms of all the elements are equal on both sides. This method of balancing equation is known as hit and trial method.

Balancing of Ionic Equations: In these equations, charge balancing of atoms on both sides of the equation, e.g., Initial Cu2+(aq.) + H2S  CuS (s) + H+ (aq) Balanced Cu2+ (aq.) + H2S  CuS (s) + 2H + (aq) We have balanced the charges. It was + 2 on LHS and we have made + 2 on RHS. Number of Hydrogen atoms, Cu and Sulphur atoms are also balanced on both sides.

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Compound

Formula

Ions involved

Sodium chloride

NaCl

Na+ and Cl–

Magnesium chloride

MgCl2

Mg2+ and Cl–

Magnesium oxide

MgO

Mg2+ and O2–

Calcium chloride

CaCl2

Ca2+ and Cl–

Calcium oxide

CaO

Ca2+ and O2–

Ammonium chloride

NH4Cl

NH4 and Cl-

Barium chloride

BaCl2

Ba2+ and Cr

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Potassium nitrate Ammonium sulphate

KNO3

K + and NO3 NH4 and

(NH4)2SO4

Cupric sulphate

CuSO4

Cupric chloride

CuCl2

2

SO 4

Cu2+ and SO 24



Cu2+ and Cl–

Important Equations Types of Reactions: S.No 1

2

3

4

5

Name & Definition

Example

In a combination reaction, two or more reactants combine to give single product. In a decomposition reaction, a single reactant breaks down into two or more simpler products. When decomposition reaction is carried out by heating, it is called thermal decomposition reaction. When decomposition reaction is carried out in the presence of sunlight, the process is called photochernical decomposition. Electrolysis: When decomposition reaction is carried out with the help of electric current, the process is called electrolysis.

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In a displacement reaction, a more reactive element displaces a less reactive element from a less compound.

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Those reactions in which the different atoms or groups of atoms are displaced by other atoms or groups of atom, i.e., two compounds exchange their ions and one of the products formed is insoluble, are said to be double displacement

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reactions. The reactions in which acid or acidic oxide reacts with the base or basic oxides to form salt and water are called neutralization reactions.

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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. H 2S04, 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) + H 20 Hydrated salts which are white in colour:

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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.

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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 CaOCI 2 is bleaching powder. By passing chlorine into dry slaked lime Ca(OH)2, bleaching powder is obtained.

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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. Substance

Acid present

1. Orange, lemon

Citric acid, ascorbic acid (vitamin C)

2. Apple

Malic acid

3. Tamarind (imli),

Tartaric acid

grape

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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)

1.

As a preservative for food

2. As a flavouring agent In the treatment of bone marrow and scurvy diseases

Acetic acid

Added to pickles to make them sour

Tartaric acid

A component of baking power (baking powder is a mixture of sodium hydrogencarbonate and tartaric acid)

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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) 1. Nitric acid present in rainwater forms nitrates in the soil which are then used by plants to obtain nitrogen.

Nitric acid

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:

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Some of the bases are listed here in Table. Oxides

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Soluble hydroxides

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Insoluble hydroxides

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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

Table lists some of the common bases and their uses. Bases

Uses

Sodium hydroxide

Ammonium hydroxide

Calcium hydroxide

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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

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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

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Sodium bicarbonate Potassium nitrate

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

Copper sulphate

1. Commonly called 'blue vitriol', used as a fungicide to kill certain germs 2. In electroplating 3. In dyeing

Potash alum

1. Used to purify water; makes suspended particles in water settle down 2. As an antiseptic 3. In dyeing

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.

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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 Gastric juice

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pH 1.0

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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

Point to Remember:                

Everything that tastes sour contains an acid. Acetic acid, citric acid, tartaric acid are a few organic acids. Sulphuric acid, nitric ac id 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.

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

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.

Key Points and Concepts  

Elements can be classified as metals and non-metals. Electronic configuration of some metals and non-metals:

Properties of Ionic Compounds: 

  

Ionic compounds are solids and are somewhat hard because of the strong force of attraction between the positive and negative ions. These compounds are generally brittle and break into pieces when pressure is applied. Ionic compounds have high melting and boiling points. Electrovalent compounds are generally soluble in water and insoluble in organic solvents such as kerosene, petrol, etc. Ionic compounds conduct electricity in the molten state.

Electronic Configuration of elements and classification as Metals and Non-metals: Electronic configuration KLMNO

Valence electrons

Metal/Nonmetal

S.No

Element

Symbol

Atomic number

1.

Hydrogen

H

1

1

1

Non-metal

2.

Helium

He

2

2

2

Non-metal

3.

Neon

Ne

10

2, 8

8

Non-metal(noble gas)

4.

Boron

B

5

2, 3

3

Non-metal

5.

Carbon

C

6

2, 4

4

Non-metal

6.

Nitrogen

N

7

2, 5

5

Non-metal

7.

Oxygen

O

8

2, 6

6

Non-metal

8.

Fluorine

F

9

2, 7

7

Non-metal

9.

Sodium

Na

11

2, 8, 1

1

Metal

10.

Magnesium

Mg

12

2, 8, 2

2

Metal

11.

Aluminium

Al

13

2, 8, 3

3

Metal

12.

Potassium

K

19

2, 8, 8, 1

1

Metal

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13.

Calcium

Ca

20

2, 8, 8, 2

2

Metal

Distinctions between metals and Nonmetals: (A) Distinctions based on physical properties Metals

Non-metals

1. Metals have lustre called metallic lustre.

Non-metals do not possess any metallic luster. Exceptions: Iodine and graphite possess metallic lustre.

2. Metals are generally electropositive

Nonmetals are generally electronegative. Exception: Hydrogen is a non-metal but it is electropositive.

3. Metals are generally good conductors of heat and electricity.

Non-metals are generally bad conductor of heat and electricity. Exception: Graphite is a non-metals but it is a good conductor of electricity. Hydrogen, though a nonmetal, is a good conductor of heat.

4. Metals are usually solids at ordinary temperature. Exception: Mercury is a metal but it is a liquid at ordinary temperature.

Nonmetals are gases or solids at ordinary temperature. Exception: Bromine is a non-metal but it is a liquid at ordinary temperature.

5. Metals are malleable and ductile, have high density, and reflect light. Exception: Sodium and potassium are metals but their densities are less than that of water

Non-metals are not malleable and ductile, have low density, and do not reflect light. Exception: Plastic sulphur is ductile.

6. Metals when hammered emit a characteristic sound called metallic sound

Nonmetals do not emit metallic sound.

(B) Distinctions based on Chemical properties Metals

Non-metals

1. Metals form positive ions. Example: Na +, Nonmetals form negative ions. Example: Cl – K +, Ca2+, Mg2+, Fe2+, etc. , S2–, N3–, etc. 2. Oxides of metals are basic in nature, i.e., Oxides of nonmetals are acidic in nature, the oxides of metals react with water to i.e., the oxides of nonmetals react with water give bases or alkalis. to give acids. Na2O + H2O  2NaOH CaO + H2O  Ca(OH)2

CO2 + H2O

 H2CO3 carbonic acid

SO2 + H2O



H2SO 3 sulphurous acid

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SO3 + H2O

3. Metals dissolve in dilute acids to produce hydrogen gas.



H 2SO 4 sulphuric acid

Nonmetals generally do not dissolve in dilute acids.

Zn + H2SO4  ZnSO4 + H2 Exceptions: Some metals do not produce hydrogen when treated with an acid under ordinary circumstances. 4. Metals in general do not combine with Nonmetals combine with hydrogen to form hydrogen. stable compounds. Exception: Some metals (Na, Ca, Li, Be, etc.) combine with hydrogen to form non volatile unstable hydrides. 5. Metallic chlorides are generally not hydrolyzed by water, or are only partially hydrolyzed. NaCl + H2O  No hydrolysis AlCl3 + 3H2O 

Chlorides of nonmetals hydrolyzed by water.

are

usually

PCl3 + 3H2O  3HCl + H3PO3 SiCl4 + 4H2O  Si(OH)4 + 4HCl

Al(OH)3 + 3HCl

Activity Series:

Important Equations Chemical Properties of Metals:

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Condition

Chemical Equation

Metals are burnt in air

Metals react with water

Metals react with acids Metals react with solutions of other metal salts

Formation of Sodium Chloride:

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(Chloride anion)

Formation of Magnesium Chloride:

(Chloride anion)

Metal Extraction: Scenario

Example

Low in the activity series

Middle in the activity series

Top in the activity series During Electrolysis

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Some important balanced equations usually asked:

Electron dot structure:

Activity Series and related metallurgy:

Minerals and Ores: Metal

Minerals

Places of occurrence in India

1. Sodium

Tincal, borax

Ladakh (Kashmir)

2. Magnesium

Dolomite, magnesite

Tamil Nadu

3. Calcium

Gypsum

Rajasthan, Tamil Nadu, Jammu and Kashmir

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4. Aluminium

Bauxite

UP, Maharashtra,'MP, Orissa

5. Copper

copper pyrites, malachite

Jharkhand, Orissa, MP

6. Zinc

Zincblende

Rajasthan

7. Manganese

Pyrolusite

Maharashtra, Karnataka and Jharkhand

8. Iron

Haematite

Jharkhand and Karnataka

Thermite Reaction:

Extraction of metals from ores:

Alloy

Composition

Uses

1. Brass

Cu = 80%, Zn = 20%

Harder than pure Cu and Zn ; used for making utensils, cartridges, etc.

2. Bronze

Cu = 90%, Sn = 10%

For making statues, medals, ships, coins, machines, etc.

3. Solder (common)

Sn = 50%, Pb = 50%

For joining metals, soldering wires,

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electronic components, etc. Al = 95.5%, Cu = 3%

4. Duralumin

In bodies of aircraft, kitchenware, automobile parts, etc

Mn = 1% Mg = 0.5%

5. Babbit metal

Sn = 90%, Sb = 7%, Cu = 3%

6. German silver

Cu = 60%, Zn = 20%, Ni = 20%

In antifriction lining For making utensils, ornaments, etc.

7. Gun metal

Cu = 60%, Sn = 10%

Gears, castings, etc.

8. Bell metal

Cu = 78%, Sn = 22%

Bells, gongs, etc.

9. Magnalium

Al = 90%, Mg = 10%

Balance beams, light instruments, etc.

10. Pewter

Sn = 75%, Pb = 25%

Cups, mugs, etc.

11. Type metal

Pb=82%, Sb = 15%, Sn = 3%

Casting type

Alloy steels Name

Composition

1. Manganese

Mn = 10 – 18%

Properties

Uses

Extremely hard, resistant to wear

Grinding machines, safes, etc. Axle and other parts of automobiles

2. Chrome vanadium

Cr = 1-10%, V = 0.15%

Highly tensile, resistance to stress and torsin

3. Nickelchromium

Ni = 1-4%, Cr = 0.5-2%

High tensile strength, hard and highly elastic

Armour plates

4. 18–8

Cr = 18%, Ni = 8%

Resistance to corrosion

Cutlery, instrument

5. Alnico

Co = 5%

Highly magnetic

Powerful permanent magnet

Point to Remember: Metals are lustrous, malleable, ductile and have high density. Metals are generally good conductors of heat and electricity. Metals form positive ions by losing electrons. All metals combine with oxygen to form metal oxides. Metal oxides are basic in nature. Metals react with chlorine to form electrovalent chlorides of metals.  Metals like Li, Na and Ca form hydrides with hydrogen.  There are metal-bearing substances below the earth's surface which are called minerals.  The minerals from which metals can be obtained profitably are called ores.      

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                

A flux is a substance which is added to the furnace charge to remove nonfusible impurities present in the ore. Flux combines with the nonfusible impurity to convert it into a fusible substance known as slag. Nonmetals are generally bad conductors of heat and electricity. All nonmetals are electronegative. The function of limestone in the extraction of iron is to provide calcium oxide (CaO) for the formation of the slag CaSiO3. Alloy steels are prepared by adding a small quantity of nickel, cobalt, chromium, tungsten, molybdenum, manganese or silicon to steel. The heating of steel to redness and then cooling it slowly is called tempering of steel. Rusting of iron is an oxidation reaction which occurs in the presence of air and water. Aluminium is the most abundant metal in the earth's crust. The process of extracting metals from their ores and refining them for use is called metallurgy. The unwanted materials present in an ore are called gangue. Froth floatation is a method used for the concentration of some ores. Calcination is the process of heating an ore strongly so that volatile impurities are removed. Roasting is the process of heating an ore in a controlled supply of air at a controlled temperature. Smelting is the process of obtaining metals from their compounds. Bauxite is an ore from which aluminium metal is commercially obtained. The most important are of aluminium is bauxite (Al2O3  2H2O).

Key Points and Concepts Function of blood vessels: S. No.

Blood vessels

Function

1

Arteries

They carry blood away from the heart to various organs of the body.

2

Veins

They collect the blood from different organs and bring it back to the heart

3

Capillaries

Exchange of material between the blood and surrounding cells takes place across the thin walls of capillaries.

Arteries and Veins: S. No. 1

Arteries They are thick walled.

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Veins They are thin walled.

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2

Arteries have no valves.

They have valves.

3

Carry oxygenated blood except pulmonary artery.

Carry deoxygenated blood except pulmonary vein.

Glands and their Secretions: S.No

Name of the Glands

Name of Secretions

1

Salivary glands

Saliva contains enzyme named ptyalin.

2

Gastric glands

Secretes gastric juice, HCI, mucus and pepsin.

3

Liver

Bile juice.

4

Intestinal glands

Instestinal juice.

5

Pancreas

Pancreatic juice which contains trypsin, lipase and amylase.

Aerobic and Anaerobic Respiration: S.No.

Aerobic Respiration

Anaerobic Respiration

1

It takes place in the presence of oxygen.

It takes place in the absence of oxygen.

2

Products obtained are CO2 and H20.

Products obtained are ethanol and CO2 or lactic acid.

3

More energy is produced.

Less energy is produced.

In human air takes the following path:

Important Equations Photosynthesis Process:

Glucose Breakdown:

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Blood Groups: Blood group

Antigen present in RBC

Antibody present plasma

Can donate to

Can receive from

A

A

B

A and AB

A, O

B

B

A

AB, B

B, O

AB

A, B

None

AB

B, AB, A, O

O

None

A and B

O, A, B, AB

O

Open and Closed Stomatal Pore:

The opening and closing of stomata depend upon the turgid or flaccid state of the guard cells. When guard cells are in turgid state the stomatal aperture opens and when guard cells are in flaccid state the stomatal aperture closes. The inner wall of guard cells (towards pore) is thick and outer wall (towards other epidermal cells) is thin. When the turgor pressure of the guard cells is increased the outer thinner wall of the guard cell is pushed out (towards the periphery) due to which a tension is created on the inner thicker wall thus pulling the inner thicker wall towards the periphery thus leading to the opening of stomatal aperture. On the contrary when the guard cells are in a flaccid state the outer thinner wall of guard cells returns to original position (moves towards pore) due to which tension on the inner wall is released which also returns to its original position and stomatal aperture gets closed again.

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Leaf Cross-Section:

Human Heart Cross-Section:

Excretory Unit of Human Kidney:

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Human Alimentary Canal:

Structure and Function of Kidney in Human: A pair of kidney is the main excretory organ in human. There are two bean shaped, dark red coloured kidney placed just below the stomach, one on each side of the mid dorsal line. The depression is called hilus from where the ureter originates and the renal artery and renal vein go in and out here. Ureter entering through hilus expands. Medula on inner side has cone like structure called renal pyramids. Each kidney contains about one million nephrons. The functional unit of the kidney is the nephron. Adrenal gland Inferior Vena cava

Renal gland Renal vein

Pelvis Medulla

Kidney

Cortex Dorsal aorta Ureter

Urinary bladder Urethra

Figure : HUMAN URINARY SYSTEM

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Respiratory System: Epiglottis Larynx Trachea

Bronchus Pleural membranes

Cut end of rib

Alveoli Pleural fluid Bronchiole

Lung Diaphragm

Figure : HUMAN RESPIRATORY SYSTEM

The mammalian respiratory system consists of nasal cavity, pharynx, larynx, trachea, bronchi, bronchioles and alveoli. Nostril: It is also called external nares. Nasal Chamber: Nasal septum divides nasal cavity into two nasal chamber by the nasal septum. Internal nares: There are posterior opening of nasal chambers that leads into pharynx. Pharynx: The pharynx provides passage to both air and food. Laryngopharynx: It is the lower part of pharynx and has a slit like aperature called glottis, which can be closed by a leaf like bilobed cartilage epiglottis, during swallowing of food bolus. Larynx: It is also called voice box or adam's apple or pomas adami. Vocal cord: In larynx, 2 pairs of vocal cord is found outer pair is false vocal cord whereas, inner pair is true vocal cord when air is forced through the larynx it cause vibration of true vocal cords and sound is produced. Trachea (Wind pipe): It is long, tubular structure which runs downward through the neck in front of oesophagus. It is supported by cartilage to prevent collapse. Primary bronchi: These are one pair of small thin walled tubular structure formed by the division of trachea. It further divides and end at alveoli. Lungs: lungs are present in thoracic cavity on either side of heart covered by pleural membrane.

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Human Excretory System:

Key Points and Concepts Control and co-ordination are the functions of the nervous system and hormones in our bodies.  The responses of the nervous system can be classified as reflex action, voluntary action and involuntary action.  Hormones produced in one part of an organism move to another part to achieve the desired effect. 

HUMAN NERVOUS SYSTEM

Central Nervous System (CNS) Brain

Spinal Cord

Peripheral Nervous System (PNS)

Voluntary Nervous System

Cranial Nerves (12 pair)

Spinal Nerves (31 pair)

Sympathetic Nervous System

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Involuntary or Autonomic Nervous System

Parasympathetic Nervous System

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Human Brain: BRAIN

Fore brain

Olfactory Lobes (2)

Cerebrum (2 Cerebral) hemispheres

Mid brain

Pons

Diencephalon Cerebral Peduncles

Hind brain Cerebellum

Medulla Oblongata

Corpora Quadrigemina

Structure of Neuron:

Neuron or nerve cell is a structural and functional unit of nervous system that is specialised to receive, conduct and transmit impulses. It is very long, sometimes reaching 90100 cm. A neuron has three parts— cell body, dendrites and axon. The term neurites is used for both dendrites and axon.

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Endocrine Glands in Human Beings:

Endocrine System: Glands 1. Pitutary (A)Adenohypophysis

Secretion (i) GH or STH (ii) TSH (iii) ACTH (iv) FSH

(v) ICSH or LH

(vi) Prolactine (vii) MSH (B) Neuro hypophysis

(i) Oxytocin (ii) Vasopressin (ADH)

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Functions Controls growth of somatic cells influences protein, carbohydrate and metabolisms Regulates the growth of thyroid gland and secretion of thyroxine Stimulates adrenal contex to grow and secrete its hormones Stimulates growth of ovarian follicles in ovary of female and controls spermatogenesis in males Stimulates ovary to produce estrogen in female and testis to produce androgens in male Controls development of mammary glands and stimulates corpus luteium to secrete progestrone Stimilate melanocytes Controls uterine contraction during parturition, stimulates lactation to increase milk secretion Controls tubular reabsorption of water in kidney, also increases blood pressure

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2. Thyroid (i) Thyroxine (ii) Thyrocalcitonin PTH

3. Parathyroid 4. Adrenal

Controls metabolism Deposit calcium over bones. Maintain blood calcium.

Mineralocorticoid (Aldosterone) Adrenaline

(A) Cortex (B) Medulla 5. Gonads (A) Testis (Cells of leydig)

Salt retention Emergency hormone Controls spermatogenesis and development of secondary sexual charaters of males

Androgens (Testosteron)

(B) Ovaries (a) Graafian follicle

(i) Estrogen

(b) Corpus luteum

(i) Progesterone

6. Thymus 7. Pineal

Development of female sexual organs. Maintain pregnancy

Thymosin Melatonin

Strenthens immune system Control skin colour.

Table: Summary of digestion enzymes of various glands with their secretions and end products of digestion in man. S.No.

Name of gland

Se cre tion

1

Salivary glands

Saliva

2

Gastric glands Gastric Juice

Site of action

Enzyme s

Food acts upon

End product

Starch

Maltose

Pepsin

Proteins

Peptones & proteoses

Renin

Casein of milk

Paracasein

Buccal cavity Salivary amylase Stomach

HCl

Stomach

—

Pepsinogen

Pepsin

3

Liver

Bile

Duodenum

—

Fats

Emulsification of fats

4

Pancreas

Pancreatic Juice

Duodenum

Amylase Trypsin Lipase

Starch & Glycogen Proteins Emulsified fats

Maltose & Isomaltose Peptones & peptides Fatty acids & glycerol

5

Intestinal glands

Intestinal Juice

Samall intestine

Erepsin Maltase Sucrase Lactase Lipase

Peptones & Peptides Maltose Sucrose Lactose Triglycerides

Amino acids Glucose Glucose & fructose Glucose & galactose Monoglycerides & fatty acid

Mucous

Large intestine

—

Lubrication of faecal matter

—

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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 10 9Nm2/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.



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

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The proportionality constant ρ 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

Current Ampere

Potential Difference Resistance Resistivity

Volt Ohm Ohm metre

Heat Electric Power

Joule Watt

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

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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 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.

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I

R 1

R 2

R 3

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 R 1, R 2, R 3 ..., etc. are combined in series, then the equivalent resistance (R) is given by, R = R 1 + R 2 + R 3 + ...

.... (i)

Derivation of mathematical expression of resistances in series combination : Let, R 1, R 2 and R 3 be the resistances connected in series, I be the current flowing through the circuit, i.e.,

passing through each resistance, and V 1, V 2 and V 3 be the potential difference across R 1, R 2 and R 3, respectively. Then, from Ohm’s law, V 1 = IR 1, V 2 = IR 2 and V 3 = IR 3

... (ii)

If, V is the potential difference across the combination of resistances then, V = V 1 + V 2 + V 3

If, R is the equivalent resistance of the circuit, then V = IR

... (iii) ... (iv)

Using Equations (i) to (iv) we can write, IR = V = V 1 + V 2 + V 3 = IR 1 + IR 2 + IR 3 or, or,

IR = I (R 1 + R 2 + R 3) R = R 1 + R 2 + R 3

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.

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R 1 I1

I

I R 2

I2

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 R 1, I1 = V/R 1

... (i)

Current passing through R 2, I2 = V/R 2

... (ii)

Current passing through R 3, I3 = V/R 3

... (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 R

=

V R 1

+

V R 2

+

V R 3

... (vi)

Cancelling common V term, one gets 1 R

=

1 R 1

+

1 R 2

+

1 R 3

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

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Symbols

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3

Plug key or switch (open)

4

Plug key or switch (closed)

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

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.

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Fleming’s Left Hand Rule:

Fleming’s left–hand left–hand rule is used to find out the direction of motion of a current– current –carrying conductor when placed in a magnetic field. This rule states as follows. Stretch out the thumb, the forefinger, 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 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

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S


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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 longitudinally (longitudinal contraction). They tend to expand laterally 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 Ele ctric Motor Motor converts electrical energy into mechanical energy. Works on the principle of Fleming’s Fleming’s left hand rule.

Generator Converts mechanical energy to electrical energy. Works on the principle of of Fleming’s right right hand rule.

Uniform Magnetic Field:

Magnetic Lines around a bar magnet:

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Magnetic Lines due to a current carrying loop:

Magnetic field in a solenoid:

Magnetic Lines around two magnets:

Magnetic Lines Produced Around a current carrying conductor:

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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. 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.

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Key Points and Concepts Law of conservation of energy: Energy can neither be created nor be destroyed, but can be transformed from one form to another.  Energy sources which are used traditionally for many years and are to deplete over a period of time are called conventional or non-renewable sources. e.g.,Coal, Petroleum, Natural gas etc.  Energy sources which do not deplete and are scarcely used by the population are called non- conventional or renewable sources of energy, e.g., Solar energy, Wind energy etc.  A good source of energy would be one, which would do a large amount of work per unit volume or mass, be easily accessible, easy to store and transport, and Economical. 

Important Equations 

The flow of energy to run a turbine in power station is: Fossil Fuels—Heat Energy —Mechanical Energy —Electrical Energy

Solar Cooker:

sun ray being reflected

Mirror

Glass sheet

A black surface absorbs more heat as compared to a white or a reflecting surface under identical conditions. Solar cookers and solar water heaters use this property in their working. Some solar cookers achieve a higher temperature by using mirrors to focus the rays of the Sun. Solar cookers are covered with a glass plate.

Hydroelectricity Generation: Reservoir Sluice gates

Power house Power lines Generator

To river

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Hydro power plants convert the potential energy of falling water into electricity. Since there are very few water-falls which could be used as a source of potential energy, hydro power plants are associated with dams. In order to produce hydel electricity, high-rise dams are constructed on the river to obstruct the flow of water and thereby collect water in larger reservoirs. The water level rises and in this process the kinetic energy of flowing water gets transformed into potential energy.

Biogas Plant: Slury

Gas outlet Manure Soil Gas tank

Soil

Fertiliser

Digester Bio-gas plant

Plant and animal products are said to be bio-mass. These fuels, however, do not produce much heat on burning and a lot of smoke is given out when they are burnt. Therefore, technological inputs to improve the efficiency of these fuels are necessary. When wood is burnt in a limited supply of oxygen, water and volatile materials present in it get removed and charcoal is left behind as the residue. Charcoal burns without flames, is comparatively smokeless and has a higher heat generation efficiency. Similarly, cow-dung, various plant materials like the residue after harvesting the crops, vegetable waste and sewage are decomposed in the absence of oxygen to give bio-gas. Since the starting material is mainly cow-dung, it is popularly known as ‘gobar-gas’.

Nuclear fission: The process of splitting of a heavy nucleus into two lighter nuclei of comparable masses (after bombardment with a energetic particle) with liberation of energy is called nuclear fission. Fission reaction of U 235 235 + n1 92U 0

 92U236 

(Unstable nucleus) 141 92 56Ba + 36Kr +

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30n1 + Q

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Ba Energy Energy Slow Neutron

235 92U

235 92U

Energy

Energy Kr

Nuclear Fusion: It has been observed that under special conditions, it is possible for the nuclei of light elements to combine and form a nucleus of a high atomic number. When two or more light nuclei moving at very high speed are fused together to form a heavy nucleus, then the process is known as nuclear fusion. 2 1H

+

Deuteron

2 1H

3 1H



Deuteron

+

1 1H

+ 4.0 MeV

Triton Proton

Energy

The triton so formed can further fuse with a third deuteron to form an –particle (Helium– nucleus). 3 1H

+ 12 H 

S.No

4 2 He

1

+ 0 n + 17.6

MeV energy

NUCLEAR FISSION

NUCLEAR FUSION

1

A heavy nucleus disintegrates to give lighter nuclei with the release of energy.

Two or more light nuclei join together to form a heavy nucleus with the release of energy

2

Nuclear fission is a chain reaction-the fission of one nucleus leading to the production of neutrons, which cause more nuclei to disintegrate.

Nuclear fusion is not a chain reaction.

3

Nuclear fission requires thermal neutrons neutronswith sufficient energy to split a nucleus. But it does not require very high temp.

Nuclear fusion requires high temperatures in the range of 106K.

Fission process can be carried out in a reactor.

Fusion cannot be carried out in any container, as the temperature is very high. Scientists hope to contain matter in a magnetic field

4

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5

Controlled nuclear fission is possible and takes place in a nuclear reactor.

Controlled nuclear fusion has so far not been achieved. Only uncontrolled fusion reactions in the thermonuclear devices such as the hydrogen bomb have been achieved.

6

Nuclear fission produces radioactive wastes, which pose the problem of waste disposal.

Nuclear fusion does not produce radioactive wastes and, therefore, does not cause pollution.

TERM-1

Fundamentals: Real Numbers

Rational no. If remainder = 0

Terminating & Non repeating

Eg:     

18 5

 3.6

Irrational no.

If remainder  0

If remainder  0

& rem. = devidend

& rem.  any devidend

Non terminating & repeating (recurring)

Eg:

1 3

 0.33.... ..

Non terminating non repeating

Eg: 0.671234…..

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. Euclid's Division Lemma: Let a and b be any two positive integers, then there exists unique integers q and r such that: a = bq + r, 0 ≤r
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Euclid's Division Algorithm: Let a and b be any two positive integers such that a>b and 'q' and ‘r’ as quotient and remainder Take a as dividend and b as divisor

a = bq + r,

0
Then every common divisor of a and b is a common divisor of b and r. 

Finding HCF of two positive integers using Euclid's Division Algorithm: Step 1: Apply Euclid's lemma to a and b: a = bq1 + r1 Step 2: If r1 = 0, then HCF = b. Step 3: If r1 ≠ 0, then again apply Euclid's lemma: b = q1r1 + r2 Step 4: If r2 = 0, then HCF = r1. Step 5: If r2 ≠ 0, then again apply Euclid's lemma till remainder rn = 0, then the divisor is HCF .

Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes, and this factorisation is unique except for the order in which the prime factors occur. Important Theorems: 1. Let p be a prime number and a be a positive integer. If p divides a2, then p divides 2. Consider two positive integers a and b, then LCM × HCF = a × b. 3. Let x be a rational number whose decimal expansion terminates. Then we can express x p in the form , where p and q are co-prime and the prime factorisation of q is of the form q n m 2 5 , where n, m are non-negative integers. p 4. Let x = be a rational number, such that the prime factorisation of q is of the form q n m 2 5 , where n, m are non-negative integers. Then x has a decimal expansion which terminates. p 5. Let x = be a rational number, such that the prime factorisation of q is not of the form q 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating.

Fundamentals: 

An algebraic equation of the form, p(x) = a0 x n + a1 x n-l + a2 x n-2 + ……… + an-1 x + an is called polynomial, provided it has no negative exponent for any variable. Where a0, a1, a2, ……an-1, an are constants (real numbers); a0≠0.

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

Remainder Theorem: 1. Remainder obtained on dividing polynomial p(x) by x – a is equal to p(a) . 2. If a polynomial p(x) is divided by (x + a) the remainder is the value of p(x) at x = –a. 3. (x – a) is a factor of polynomial p(x) if p (a) = 0 4. (x + a) is a factor of polynomial p(x) if p (–a) = 0 5. (x – a)(x – b) is a factor of polynomial p(x), if p(a) = 0 and p(b) = 0.



Degree of polynomial: n is called the degree (highest power of variable x ). If n = 1 then polynomial is called linear polynomial. General form:

ax + b = 0

(a≠ 0)

If n = 2 then polynomial is called quadratic polynomial. General form:

ax 2 + bx + c = 0

(a≠0)

If n = 3 then polynomial is called cubic polynomial. General form:

ax 3 + bx 2 + cx + d = 0 (a≠0)

Zeroes of Polynomial: For polynomial p(x), the value of x for which p(x) = 0, is called zero(es) of polynomial. Linear Polynomial can have at most 1 root. Quadratic Polynomial can have at most 2 roots. Cubic Polynomial can have at most 3 roots.

Types of Polynomials: (A)

Based on degree: If degree of polynomial is Examples 1.

One

Linear

2.

Two

Quadratic

2x2 –7, x2+y 2 –2xy, x2 +1+ 3y

3.

Three

Cubic

x3 + 3x2 –7x+8, 2x2+5x3+7,

4.

Four

bi-quadratic

x4 + y 4 + 2x2 y 2, x4 + 3,…

x + 3, y – x + 2,

3 x –3

1

3

(B) Based on Terms: If number of terms in polynomial is

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Examples 7x, 5x9,

7

x16, xy, ……

1.

One

Monomial

2.

Two

Binomial

2 + 7y 6, y 3 + x14, 7 + 5x9,…

3.

Three

Trinomial

x3 –2x + y, x31+y 32+ z33,…..

3

Relationship between zeroes and coefficient of polynomial: b (constant term) Zero of linear polynomial ax + b = 0 is given by x = 



a

=

(coeffient of x)

.

If α and β are zeroes of the quadratic polynomial ax 2 + bx + c = 0, then Sum of zeroes

α+β =

Products of zeroes

αβ =

c a

b

=

a =

(coeffient of x) (coeffient of x 2 )

(constant term) (coeffient of x 2 )

If α , β and γ are the zeroes of the cubic polynomial ax 3 + bx 2 + cx + d = 0, then

b

Sum of zeroes

α+β+ γ =

Product of zeroes

αβ + β γ + γα =

αβ γ =

a

=

(coeffient of x 2 ) (coeffient of x3 ) c a

=

(constanat of x ) (coeffient of x3 )

d (constant term) a

=

(coeffient of x3 )

Division algorithm for polynomials: If p(x) and g(x) are two polynomials with g(x) ≠ 0, then we can find q(x) and r(x) such that p(x) = g(x) × q(x) + r(x), In simple words: Dividend = Divisor

×

Quotient + Remainder

Tips: 1. Graph of linear polynomial is a straight line, while graph of quadratic equation is a parabola. 2. Degree of polynomial = maximum number of zeroes of polynomial. 3. If remainder r(x) = 0, then g(x) is a factor of p(x). NCERT Solutions

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4. To form quadratic polynomial if sum and product of z eroes are given p(x) = x 2 - (Sum of zeroes) x + (Product of zeroes)

Fundamentals:           

The general form of linear equation in one variable is: ax + b = 0, a≠0. The graph of a linear equation in two variables is a straight line A linear equation in two variables has infinitely many solutions Slope of the line ax + by + c = 0 is – a/b Equation of x-axis is y = 0 and equation of y-axis is x = 0 The graph of the line x = a is parallel to y-axis. The graph of the line y = b is parallel to x-axis. Every point on the graph of a linear equation in two variables is a solution of the equation. The general form of linear equation in two variables is: ax + by + c = 0. where a, b, c are real coefficients; a ≠0,b≠0 Two linear equations which are in the same two variables x and y simultaneously are called pair of linear equation in two variables. The general form of pair of linear equation in two variables is : a1 x + b1y + c1 = 0 and

a2 x + b2y +c2 = 0

Where a1, b1, c1, a2, b2 ,c2 are all real coefficients  A pair of values of x and y satisfying each one of the equations is called solution of the system.

Graphical representation of pair of linear equations: Let the system of pair of linear equations be

and

a1 x + b1y + c1 = 0

.... (1)

a2 x + b2y +c2 = 0

.... (2)

We know that given two lines in a plane, only one of the following three possibilities can happen (i) The two lines will intersect at one point. (ii) The two lines will not intersect, however far they are extended, i.e., they are parallel. (iii) The two lines are coincident lines.

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Y

1

Y

Y

1 and 2

2

1 2

P() 0

' 0

0

Y'

Graphical and Algebraic Interpretation: Pair of Linear Equations

Algebraic Condition

a1 a2 a2 x + b2y + c1 = 0 a2 x + b2y + c2 =0

a1 a2 a1 a2

= =

≠

b1 b2 b1 b2

b1 b2 =

≠

c1 c2 c1 c2

Graphical Interpretation

Algebraic Interpretation

Consistency

Intersecting lines

Unique solution

Consistency

Coincident lines

Infinitely many solutions

Dependent Consistent

Parallel lines

No solution

Inconsistent

Methods of Solving Pair of Linear Equations: 1. 2. 3. 4. 5.

Graphical Method Algebraic Method Substitution Method Elimination Method Cross-Multiplication method

Graphical Method: Step 1: Plot the both linear equations on the same graph. Step 2: Find the intersecting point on graph, if the lines are intersecting. Step 3: Intersecting point is the required solution. Step 4: In step 2, if the lines are coincident, then there are infinitely many solutions-each point on the line being a solution. Step 5: In step 2, if the lines are parallel, then the pair of equations has no solution.

Substitution Method: Step 1: From one equation, find the value of one variable in terms of other v ariable. Step 2: Substitute the value of variable obtained in step 1 in the other equation, you will get the equation in one variable.

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Step 3: Solve the equation in one variable and find the value of variable. Step 4: Substitute the value of the variable so obtained in step 3 in any equation, you will get equation in unknown variable. Step 5: Solve this equation in one variable and find the value of this variable.

Elimination Method: Step 1: First multiply both the equations by some suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal. Step 2: Then add or subtract one equation from other so that one variable gets eliminated and resultant equation will become an equation in one variable. Step 3: Solve the equation in one variable and find the value of the variable. Step 4: Substitute the value of the variable so obtained in step 3 in any equation, you will get equation in unknown variable. Step 5: Solve this equation in one variable and find the value of this variable now.

Cross Multiplication Method: For two pair of linear equations, a1 x + b1y + c1 = 0 and

a2 x + b2y +c2 = 0, make an array as below:

From this,

you will get: x 

b1c2  b2c1 a1b2  a2b1

and y =

c1a2  c2a1 a1b2  a2b1

, a1b2  a2b1 ≠ 0

Equations Reducible to A Pair of Linear Equations in Two Variables: There are several situations which can be mathematically represented by two equations that are not linear to start with. But we alter them so that they are reduced to a pair of linear equations.

Fundamentals: Similar figures: Two figures of same shape are said to be similar if: 1. Their corresponding angles are equal. 2. Their corresponding sides are proportional. Examples: a) All circles

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b) All squares c) All equilateral triangles d) All congruent triangles.

Similar triangle and their properties: Two triangles are said to be similar, if their (i)

Corresponding angles are equal and,

(ii)

Corresponding sides are proportional.

Two triangles ABC and DEF are similar, if (i)

 A = D, B = E, C = F and, AB

(ii)

DE

BC

=

EF

AC

=

DF D A

B

C E

F

Theorem 1: If two triangles are equiangular, prove that the ratio of the corresponding sides is same as the ratio of the corresponding medians. Given: Two triangles ABC and DEF in which  A = D, B = E, C = F, AP and DQ are their medians. D A

B

P

C

E

Figure (i) To Prove:

BC EF

=

Q

F

Figure (ii)

AP DQ

Proof: Since equiangular triangles are similar.   ABC ~ DEF 

AB DE

=

BC EF



AB DE

=

2BP 2EQ

 P and Q are mid  po int s of BC  and EF respectively    BC  2BP and EF  2EQ 

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AB



DE

BP

=

EQ

.... (ii)

Now, in  ABP and DFQ, we have AB DE

and,

BP

=

EQ

B = E

[From (ii)] [Given]

So, by SAS-criterion of similarity, we have  ABP ~ DEQ 

AB DE

=

AP DQ

.... (iii)

From (i) and (iii), we get BC EF

=

AP DQ

Hence, the ratio of the corresponding sides is same as the ratio of corresponding medians.

Statement of Theorems 1. Basic Proportionality Theorem: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. 2. Converse of Basic Proportionality Theorem: If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side. 3. Ratio of The Areas of Two Similar Triangles: The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides. 4. Pythagoras Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 5. Converse of Pythagoras Theorem: In a triangle, if the square of one side is equal to sum of the squares of the other two sides, the angle opposite to the first side is a right angle. 6. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.

Criterion for Similarity of Two Triangles SSS Similarity: If the corresponding sides of two triangles are proportional, then triangles are similar.  AAA Similarity: If the corresponding angles of two triangles are equal, then triangles are similar.  SAS Similarity: If the pair of one corresponding sides of two triangles are proportional and the included angles are equal, then triangles are similar. 

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Proof of Theorems: Basic Proportionality Theorem (Thales Theorem): Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Given: A ∆ ABC and line ‘' parallel to BC intersect AB at D and AC at E

AD

To Prove:

DB

AE

=

Construction: Join BE and CD. Draw EL

EC

 AB and DM  AC .

Proof: We know that areas of the triangles on the same base and between same parallel lines are equal, hence we have

area ( BD E)  area ( CDE )

…. (i)

1 ×AD×EL area (Δ ADE) AD 2 = = 1 area (Δ BDE) DB ×DB×EL 2

.… (ii)

Now, we have

Again, we have 1

×AE×DM AE 2 = = 1 area (Δ CDE) EC ×EC×DM 2

area (Δ ADE)

… (iii)

Put value from (i) in (ii), we have area (Δ ADE) area (Δ CDE)

=

AD DB

…. (iv)

On comparing equation (ii) and (iii), we get NCERT Solutions

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AD DB

=

AE

Hence Proved.

EC

Corollary: (i)

AB DB

(iv)



DB AB

AC EC



EC AC

(ii)

(v)

DB AD AD AB



EC



AE

AC

(iii)

AB AD



AC AE

AC

Converse of Basic Proportionality Theorem: Statement: If a line divides any two sides of triangles in the same ratio, the line must be parallel to the third side. Given: A ∆ ABC and line ‘' intersecting the sides AB at D and AC at E such that:

AD DB

=

AE EC

To Prove:  BC. Proof: Let us suppose that the line l is not parallel to BC. Then through D, there must be any other line which must be parallel to BC. Let DF BC, such that E ≠ F. Since,

DF BC AD DB

AD DB

=

=

AF FC

AE EC

(By supposition) … (i) (Basic Proportionality Theorem)

… (ii)

(Given)

Comparing (i) and (ii), we get

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AF FC

=

AE EC

Adding 1 to both sides, we get AF FC

+1 =



AF+FC



AC



1

FC

FC

FC

 DF

=

=

AE

+1

EC

=

AE+EC EC

AC EC 1 EC

FC=EC This shows that E and F must coincide, but it contradicts our supposition that E ≠ F and BC. Hence, there is one and only line, DF BC, i.e. 

BC

Hence Proved.

Ratio of the Areas of Two Similar Triangles Statement: The ratio of the areas of the two similar triangles is equal to the ratio of squares of their corresponding sides. Given: To prove:

∆ ABC ~ ∆ DEF

area (Δ ABC) area (Δ DEF)

2

=

BC

2

EF

2

=

AB DE

2

2

=

AC

2

DF

Construction: Draw AG  BC and DH  EF .

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1

×BC×AG BC 2 = = 1 area (Δ DEF) EF ×EF×DH 2

area (Δ ABC)

Proof:



AG

… (i)

DH

( area of Δ = 1 ×base×height ) 2 Now in triangle ABG and DEH , we have

B  E  AGB   DHE ABG DEG

Therefore,

AB

Hence,

DE AB



= =

DE

AG

(since, ABC

DEF) 0

(each 90 ) (by AA criterion)

... (ii) (Using property of similar triangles)

DH BC

... (iii) (since, ABC

EF

DEF)

Comparing (ii) and (iii), we get

AG DH

=

BC

… (iv)

EF

Using (i) and (iv), we get

area (Δ ABC) area (Δ DEF) BC EF

=

2

=

AB DE

BC

=

2

EF

2

=

AB DE

2

2

=

AC

… (v)

2

DF

AC

... (vi) (since, ABC

DF

DEF)

Using (v) and (vi), we get area (Δ ABC) area (Δ DEF)

2

=

BC

2

EF

2

=

AB DE

2

2

=

AC

2

DF

Hence Proved. Corollary: 1. The ratio of the areas of the two similar triangles is equal to the ratio of squares of their corresponding altitudes. 2. The ratio of the areas of the two similar triangles is equal to the ratio of squares of their corresponding medians.

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Pythagoras Theorem (Baudhayan Theorem) Statement: In a right triangle, the square of one side (longest side, i.e. hypotenuse) is equal to the sum of squares of other two sides (i.e. base and perpendicular). Given: ∆ ABC is right angled at B. To Prove:

AC2 = AB2 + BC2

Construction: Draw BD  AC

Proof: Taking ∆ ADB and ∆ ABC

Therefore, Hence,



 B =  ADB A=A ADB ABC AD AB

=

(each 900 )

(common)

(by AA criterion)

AD AB

2

AB = AD×AC

... (i)

Now taking ΔCDB and ΔCBA

Therefore, Hence,

 B =  BDC C=C CDB CBA CD BC

=

(each 900 ) (common) (by AA criterion)

BC AC



BC2 = CD×AC

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... (ii)

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Adding (i) and (ii), we get

  

AB2 +BC2 = AD×AC + CD×AC AB2 +BC2 = AC  (AD + CD) AB +BC = AC  AC 2

2

2

2

AC = AB +BC

2

Hence Proved

Converse of Pythagoras Theorem Statement: In a right triangle, if the square of one side (longest side, i.e. hypotenuse) is equal to the sum of squares of other two sides (i.e. base and perpendicular), then the angle opposite to the first side is the right angle. Given:

AC2 = AB2 + BC2

… (i)

To Prove: ∆ ABC is right angled at B. Construction: Draw right ∆ PQR such that AB = PQ, BC = QR and  Q = 90°.

Proof: Using Pythagoras theorem in ∆ PQR, we get 2

2

PR 2 = PQ + QR By construction,

… (ii)

AB = PQ BC = QR, substituting these values in (ii), we get PR 2 = AB2 +BC 2

... (iii)

comparing (i) and (iii), we get AC2 = PR 2



AC = PR

... (iv)

In ABC and PQR

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AB = PR

(by construction)

BC = QR

(by construction)

AC = PR

(Proved above in (iv))

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 

ABC = PQR B=Q  Q = 900  B = 900

But Hence,

(by SSS congruence rule) (by cpct) (by construction)

ABC is right angled at B.



Hence Proved.

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

=

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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

= =

= =

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BC AC AB AC BC AB AC BC

= =

= =

Perpendicular Hypotenuse Base Hypotenuse Perpendicular Base Hypotenuse Perpendicular

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5. Secant A

=

6. Cotangent A

=

Hypotenuse Adjacent side to angle A Adjacent side to angle A Opposite side to angle A

= =

AC AB AB BC

= =

Hypotenuse Base Base Perpendicular

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.

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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

(P)

HAR

(H)

HAR

(H)

BOLE

(B)

Similarly: cosec

sec

cot

HAR

(H)

HAR

(H)

BOLE

(B)

PADNIT

(P)

BADRI

(B)

PRASAD

(P)

Relation between Trigonometric Ratios: sin θ =

cos θ =

tan θ =

tan θ =

1 cosec θ 1 sec θ 1 cot θ sin θ cot θ

OR

cosec θ =

OR

sec θ =

OR

cot θ =

OR

cot θ =

1 sin θ

1 cosθ

1 tan θ cosθ sin θ

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?

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1. First of all learn only sin row. If you can't learn then follow the step to find sin θ row

Sin

0°

30°

45°

60°

90°

0

1

2

3

4

4

4

4

4

4

0

1/2

1/ 2

3/2

1

2. For cos θ row, write all the values of sin θ row in inverse order, i.e. from right to left. sin θ tan θ = 3. For cot θ 4. For

cosec θ =

5. For

sec θ =

6. For

cot θ =

1 sin θ 1 cos θ cos θ sin θ

Trigonometric Ratios of Complementary Angles: In ∆ ABC ,

 B = 90°, Let  A = θ, hence  C = 90° - θ.

Thus angles θ and (90° - θ) are complementary angles.

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Fundamental Trigonometric Identities:

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. 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 =

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lower limit  upper limit 2

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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

f1 , f 2 , f3 , ....... f n , 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 

 f x

i i

i 1 n

 f i

i 1

2. Assumed Mean Method: The formula used in the assumed mean method is given as below: n

 f d i

Mean ( x ) = a+

i

i 1 n

 f i

i 1

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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: ( x ) = a +

  fi ui     h , f   i 

Where a is any arbitrary value, chosen as assumed mean (somewhere in the middle of xi ). h = class-size and

ui



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 n

=

1 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: th

 n  1  , observation is the median. Then,    2  Step 3: If the total frequency n is even: th

n  Then, mean of and   1  , observations the median. 2 2  nth

Condition II: When the data is continuous and in the form of frequency distribution:

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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.



 h 2 f  f  f  1 0 2

Mode  l   Where,

f1  f 0

l = lower limit of the modal class

h = class-size f 1 = 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

on 2 the y-axis. Locate corresponding point on the ogive, x -coordinate of that point determines the median of the data.

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TERM-2

Key Points and Concepts: Covalent and Ionic Compounds: S.No

Covalent Compounds

1.

They are readily soluble in organic solvent.

2.

They do not ionise.

Ionic Compounds They are not soluble in organic solvent. They ionise in organic medium.

3.

They are bad conductors of heat and electricity.

They are good conductors of heat and electricity.

4.

They have weak force of attraction between the molecules.

They have strong force of attraction between the molecules.

Formation of molecules: Hydrogen Molecule: When two atoms of hydrogen share one electron each, a single covalent bond is formed as shown below:

Fig. Formation of Hydrogen

Oxygen Molecule: When two oxygen atoms share two electrons each to complete their octet, double covalent bond is formed.

Fig. Formation of Oxygen

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Chlorine Molecule: Chlorine has 7 valence electrons. It can share one electron with other chlorine atom to form Cl2.

Fig. Formation of Chlorine

Nitrogen: Nitrogen has five valence electrons. It needs three more electrons to complete its octet. It shares three electrons with other atom of nitrogen to form triple covalent bond.

Fig. Formation of Nitrogen

Ammonia: Nitrogen has five valence electrons. It shares one electron with each of the three hydrogen atoms to form NH3.

Fig. Formation of Ammonia

Methane: Carbon has four valence electrons. It needs four electrons to complete its octet. It shares four electrons with four hydrogen atoms and forms four single covalent bonds.

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Fig. Formation of Methane

Water: In formation of H2O, each hydrogen atoms shares one electron with oxygen atom so that oxygen completes its octet and hydrogen acquires nearest nobles gas configuration.

Fig. Formation of Water

Carbon dioxide: Carbon has four valence electron. It shares two electron with one of the oxygen and two electrons with other atom of oxygen to form double covalent bond.

Fig. Formation of Carbon dioxide

Classification of Hydrocarbons:

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IUPAC Nomenclature: Some Common Alkanes: ALKANES

ALKYLS

methane

CH4

methyl

CH3

ethane

C2H6

ethyl

C2H5

propane

C3H8

propyl

C3H7

butane

C4H10

butyl

C4H9

pentane

C5H12

propyl

C5H11

hexane

C6H14

hexyl

C6H13

heptane

C7H16

heptyl

C7H15

octane

C8H18

octyl

C8H17

nonane

C9H20

nonyl

C10H19

decane

C10H22

decyl

C10H21

Homologous Series of Alkene: Alkyl group

Derived from Alkane

Name Of Alkyl Group

-CH3

Methane

methyl

-C2H5

Ethane

ethyl

-C3H7

Propane

propyl

Homologous Series of Alkene: No. of carbon atoms 2

Name of the Alkene

Molecular formula

Ethene

C2H4

3

Propene

C3H6

4

Butene

C4H8

5

Pentene

C5H10

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Homologous Series of Alkyne: No. of carbon atoms

Name of the Alkene

2

Ethyne

Molecular formula C2H2

3

Propyne

C3H4

4

Butyne

C4H6

5

Pentyne

C5H8

Graphite, Diamond and Fullerene:

Diamond: 1. It is hardest substance known and its density is 3.5 g/ml. 2. Its crystals are octahedral, colourless and transparent 3. In diamond, each carbon atom is covalently bonded to four other carbon atoms along four corners of regular tetrahedron. This pattern extends in three dimensions. Diamond is hard due to strong covalent bonds present in it. 4. Diamond is non-conductor of electricity 5. The standard heat of formation of diamond is 29 kJ mol –1. Graphite: 1. Graphite is soft and slippery with density of 2.3 g/ml 2. It is black coloured, opaque and has hexagonal crystals. 3. In graphite, carbon atoms are bonded together in flat layers by strong covalent bonds in a regular hexagon. These layers are held together by much weaker van der Waal’s forces, therefore the crystals of graphite are soft and slippery. 4. Graphite is conductor of electricity. 5. It is thermodynamically most stable.

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Common Functional Group:

Some Common Alcohols: Methanol (CH3OH) 1

CH3OH

Ethanol (C2H5OH) 2

CH3CH2OH

Propanol (C3H7OH) 3

CH3CH2CH2OH

Some Common Aldehydes: Formaldehyde (CH2O) 1

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HCOH

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Ethanal (C2H4O) 2

CH3COH

Some Common Carboxylic Acids: Methanoic Acid (CH2O2) 1

HCOOH

Ethanoic Acid (C2H4O2) 2

3

CH3COOH

Propanoic Acid (C3H6O2) CH3CH2COOH

Some Common Ketones: 2-Propanone (C3H6O)

1

CH3COCH3

2-Butanone (C4H8O) 2

3

CH3CH2COCH3

2-Pentanone (C 5H10O) CH3CH2CH2COCH3

Point to Remember: 1. 2. 3.

Carbon always forms covalent bonds. Carbon is present in all substances of animal and vegetable origin The ability of carbon to unite with an indefinite number of carbon atoms in straight, branched or cyclic chains is known as catenation. 4. Caron and hydrogen combine together indifferent proportions to form a large number of compounds called hydrocarbons. 5. There are two types of hydrocarbons-saturated and unsaturated

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6. Alkanes are represented represented by the general general formula formula C nH2n+2 7. Alkenes are are represented represented by the general general formula formula CnH2n 8. Alkynes are represented represented by the general general formula formula C nH2n-2 9. Organic compounds having the same functional group and common properties, but differing in molecular formula from the next member by one CH2 group, form a homologous series and such compounds are called homologues. 10. Compounds with the same molecular formula but different structural formulae are known as isomers. 11. The decomposition of alkanes on heating in the absence of oxygen is known as cracking. 12. Methane is prepared by heating a mixture of sodium acetate and soda lime.

Key Points and Concepts: Characteristics Characteris tics of Dobereiner Dobereiner's 's triads: 1. 2.

Properties of elements in each triad were similar. Dobereiner showed that when three elements in a triad were written in the order of their increasing atomic masses, the atomic mass of the middle element was roughly the average of the atomic masses of the other two elements. elements.

Dobereiner's triad: Triad 1

Triad 2

Triad 3

Lithium (Li)

Calcium (Ca)

Chlorine (Cl)

Sodium (Na)

Strontium (Sr)

Bromine (Br)

Potassium (K)

Barium (Ba)

Iodine (I)

Limitation of Dobereiner's triads: Dobereiner triads: Dobereiner could identify only three triads. He was not able to prepare triads of all the known elements. Newlands' Octave: John Octave: John Newlands arranged the elements in order of their increasing atomic mass. It states that whenever the elements are arranged in increasing order of their atomic mass, the properties of the eighth element are a kind of repetition of the first, just like the notes of music.

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Limitations of Newlands law of octaves:  



The law was applicable to elements elements upto calcium (Ca) only. It contained only 56 elements. Further it was assumed by Newlands that only 56 elements existed in nature and no more elements are going to be discovered in the future. In order to fit these elements into the table, Newlands adjusted two elements in the same slot and also put some unlike elements elements under the same note. For example, cobalt and nickel are in the same slot and these are placed in the same column as fluorine, chlorine and bromine which have very different properties than these elements. Iron, which resembles cobalt and nickel in properties, has been placed differently away from these elements.

Characteristics Characteris tics of Mendeleev's periodic table:   

Mendeleev arranged all the 63 known elements in increasing order of their atomic masses. The table consists of vertical columns called 'groups' and horizontal rows called 'periods'. The elements with similar physical and chemical properties came under same groups.

Limitations of Mendeleev's periodic table: He could not assign a correct position to hydrogen in the periodic table. Positions of isotopes of all the elements were not certain according to Mendeleev's Mendeleev's periodic table. regular manner manner in going from from one element element to the  Atomic masses did not increase in a regular next. So it was possible to predict how many elements could be discovered between the two elements.  

The Modern Periodic table:   



The Modern Periodic table table consists of 18 groups and 7 periods. Elements present in any one group have the same number of valence electrons. Also, the number of shells increases as we go down the group. Elements present in any one period, contain the same number of shells. Also, with increase in atomic number by one unit on moving from left to right, the valence shell electron increases by one unit. Each period marks a new electronic shell getting filled.

Modern Periodic Law: ‘‘Properties ‘‘Properties of elements are periodic functions of their atomic numbers, i.e., the number of protons or electrons electrons present in the neutral atom of an element.’’ Long form of Periodic Table: Arranged Table: Arranged in increasing order of their their atomic numbers. numbers. The prediction of properties elements and their compounds can be made with precision. All drawbacks of Mendeleev's Periodic Table vanish when the elements are arranged on the basis of increasing atomic numbers. Main Features of the Long Form of the Periodic Table: 1. It shows arrangement of elements based on modern periodic law.

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2. There are 18 vertical columns known as groups. 3. There are 7 horizontal rows known as periods. 4. Elements Elements having similar outer electronic configurations, i.e., having same valence electrons have been placed in same groups

Properties of Metals: 1. They are malleable. 2. They are ductile. 3. They are good conductors of heat and electricity. 4. They have generally 1 to 3 valence electrons. 5. They have the same or less number of electrons in their outermost shell than the number of shells. 6. They are mostly solids.

Properties of non-metals: 1. 2. 3. 4.

They exist in solid, liquid or gaseous state. Non-metals are generally brittle. They are non-conductors. non-conductors. They have 4 to 8 valence electrons. electrons.

Periodic Table:

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Key Points and Concepts: Sexual and Asexual Reproduction: S. No.

Sexual Reproduction

Asexual Reproduction

2.

It produces new organism from two It produces new organism from a single parents. parent. It involves sex cells or gametes. It does not involves sex cells or gametes.

3.

Offspring's Offspring's are not identical to the Offspring7 s are identical to the parents.

1.

Why avoid avoid frequent frequent pregnancies? pregnancies? 1. 2.

It has adverse effect on the health of a woman. It increases the rate of the population of our country.

Method of Contracep Contraception: tion: 1.

2. 3.

Barrier method: In this method, a device is used to prevent the entry of sperms in the female genital tract during sexual intercourse. Example: Condom, diaphragm and cervical cap. Chemical method: It involves the use of specific drugs by females. Example: Oral pills, vaginal pills, OC. Surgical method : Surgical removal or ligation of vas deferens deferens in males and the fallopian tube in females, thereby preventing production of male and female gametes.

STD (Sexually Transmitted Diseases):  

Disease caused by virus— virus— AIDS, Genital Genital Warts. Disease caused by bacteria— bacteria—Gonorrhea, Gonorrhea, Syphilis.

Prevention of Diseases:    

By use of devices made of plastic, metal or a combination of the two which is inserted into uterus. By using contraceptive devices. By educating people and maintaining hygiene. By avoiding multiple sexes.

Advantages of Vegetative Vegetative Propagation: Propagation: 1. 2.

Plants raised by vegetative reproduction can bear flowers and fruits earlier than those produced from seeds. This process helps those plants to propagate that have lost the capacity to reproduce.

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DNA Copying: 



Cells use chemical reactions to build copies of their DNA. This creates two copies of the DNA in a reproducing cell. cell. DNA copying is accompanied accompanied by the creation of an additional cellular apparatus to facilitate the DNA copies to separate with its own cellular apparatus. DNA copying gives rise to some inbuilt tendency for variation during reproduction which is the basis for evolution.

Longitudinal Section of a flower:

Sepals: Sepals Sepals: Sepals are green outermost leaf-like floral organs which protect the flower in the bud stage. In the mature stage they provide support to other floral organs. Petals: Petals Petals: Petals are coloured accessory floral organs which lie above the sepals. Petals attract insects to flowers for pollination. Stamens: Stamens Stamens: Stamens are the male reproductive organs of the flowers. Each Stamen has a slender stalk called filament.

Regeneration in Planaria:

1. It is the ability of an organism to replace its lost body parts. 2. A special case of regeneration regeneration is fragmentation, fragmentation, in which a parent multicelluar multicelluar organism organism on maturing breaks up naturally to produce two or more daughter organisms. 3. Among plants, plants, filamentous filamentous algae like like spirogyra reproduce reproduce by this this method. 4. Hydra, Planaria & Sponges exhibit regeneration. regeneration.

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Budding in Hydra:

1. In budding a small part of the body of the parent organism grows out as a ''bud'' which then detaches and become new organism. 2. The nucleus of the parent divides and one of the daughter nuclei passes into the bud. 3. The bud detaches itself from the parent body & becomes a new individual after growing to full size. 4. In plants, budding takes place in yeast and in animals budding is seen in hydra & sponges.

Germinated Seed:

Germination of pollen on stigma:

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Pollination is followed by fertilisation in plants.     

 

After reaching the stigma, the pollen grain develops the pollen tube. This pollen tube grows through the style and reaches the ovary where ovules are located. The pollen tube enters the ovule through a small opening called micropyle, where it releases two male gametes into the embryo sac. One male gamete fuses with the egg contained in the embryo sac of the ovule; and this fusion of male and female gametes is called syngamy and its product is the zygote. The other male gamete fuses with the two polar nuclei and this process is called triple fusion, where three nuclei are involved in the fusion process, one male gamete and two polar nuclei. The process of double fertilisation occurs inside each embryo sac, in which two fusions, syngamy and triple fusion take place. After fertilisation, ovary develops into the fruit and ovules develop into the seeds.

Human Female Reproductive System:

The human female reproductive system consists of the following organs: Ovaries: Ovaries are a pair of small and oval-shaped organs, located in the abdominal cavity near the kidney. Ovaries are the female primary reproductive organs which perform dual functions of production of female gamete or ovum and the secretion of female sex hormones, estrogen and progesterone. Fallopian Tube or Oviduct - are a pair of long convoluted tubes that carry ovun or eggs from the ovary to the uterus. The fallopian tube has a funnel-shaped opening near the ovary. These tubes from both the sides open into a muscular structure, the Uterus. Uterus or womb - is a hollow, pear-shaped organ within which the embryo develops. Its upper portion is broader, while its lower portion is narrower, called cervix.

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Vagina - The cervix opens into the vagina which is a tubular structure and also called ''birth canal'' or canal for menstrual flow. Vagina receives sperms from the male and also serves as the passage through which the fully developed foetus is born.

Human Male Reproductive System:

The human male reproductive system consists of the following organs: A pair of testis lies in a small sac-like muscular structure outside the abdominal cavity called Scrotum. The function of testis is to produce sperm and male sex hormone called testosterone. The scrotum provides the optimal temperature for formation of sperms. Epididymis is a coiled tube-like structure firmly attached to the testis and serves as the storehouse of sperms. Inside the epididymis, sperms become mature and develops motility. Vas deferens: The sperms are carried by a long tube called vas deferens or sperm duct into organs called seminal vesicles, where the sperms get nourished and stored. Seminal vesicle is a glandular structure which joins vas deferens to form ejacuatory duct. Ejaculatory duct enters prostate gland and joins urethra to form common urino-genital duct. Penis is a copulatory organ at the tip of which urinogenital duct opens.

Key Points and Concepts: Heredity is the transmission of traits from one generation to the next with the help of genes.  Variation is the degree of difference found in morphological, physiological & other traits found among individuals belonging to the same family race & species.  Most of the variation are due to mutations caused by error in DNA replication.  Variation appear during reproduction whether organisms are multiplying asexually or sexually. Each generation provides the next generation with a common basic body design and some subtle changes or variations 

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The variations accumulate and pass on to more and more individuals with each generation. The diversity is small in case of asexually reproducing organism as it is caused by only errors of DNA copying mechanism.

 

Acquired and Inherited Traits: S. No.

Acquired Traits

1.

The traits develop during the lifetime of an individual.

The traits are obtained from the parents.

2.

They are somatic variations and die with the death of the individual. e.g., Muscular body of wrestler.

They are genetic variations which passes on to the next generation. e.g., Fused and free ear lobes.

      

Inherited Traits

Genetic drift and natural selection could lead to the rise of new species. In case of monohybrid cross with pure variety of plants, the phenotypic ratio obtained in F2 generation is 3 : 1. In case of di-hybrid cross involving two pairs of contrasting characters, the phenotypic ratio obtained in F2 generation is 9 : 3 : 1 : 1. Genes carry information for producing proteins, which in turn control the various body characteristics. Humans have 22 pairs of autosomes and one pair of sex chromosomes. Females have similar sex chromosomes XX, whereas males have an imperfect pair i.e., XY. All eggs carry X chromosome. The sex of the child depends on whether the egg fuses with the sperm carrying X chromosome (resulting in a girl) or with the sperm carrying Y chromosome (resulting in a boy).

Sex Determination: It is a process by which the sex of a person is determined. Genetics is involved in the determination of the sex of a person, which is explained as follows:      

A male has one X chromosome and one Y chromosome, i.e., half of the male gamete or sperms will have X chromosome and the other half will have Y chromosome. A female has two X chromosomes, i.e., all the female gametes or ova will have only X chromosomes. Sex of a child depends on what happens at fertilisation. If a sperm carrying X chromosome fertilises an ovum which carries X chromosome, then the child born will be a girl. If a sperm carrying Y chromosome fertilises an ovum which carries X chromosome, then the child born will be a boy. Thus, the sperm determines the sex of the child. Sex determination is also controlled by the environmental factors in some animals.

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

For example, in some reptiles like turtle, high incubation temperature leads to the development of female offsprings, while in case of lizard, high incubation temperature results in male offsprings. Father's Sperms

Mother's Ova Gametes

x

Y

Zygote Offsprings

xx

xy

Female Child (Girl)

Male Child (Boy)

Determination of Sex in Humans

Mendel's law of Dominance: According to this law, if a cross is made between plants with contrasting pair of charactess, the character that appears in the first generation is dominant and the other is recessive. Parents

× Red (RR) Dominant

White (rr) Recessive F1 generation Red (Rr)

Mendel's monohybrid cross between a homozygous red flowered (RR) and a homozygous white flowered (rr) plant to show that red colour is dominant over white (recessive)

Determination of sex of a child:

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Inheritance of two characters over two generations by making a cross between round and yellow seeded plant (RRYY) with wrinkled green seeded plant (rryy).

Work Done by Mendel: Ratio of tall to short in F2 generation is 3:1

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Key Points and Concepts: 1

1

1

v

u

f

 Mirror Formula :    Lens Formula :

1 v

1

1

u

f

 

 Linear Magnification:

m= m=-

v u v u

or

h h1

or -

h h1

(in lens) (in mirror)

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

= µ

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.

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Sign Convention in Mirror

Sign Convention in Lens:

1



Power of a lens P =



The unit for power is Dioptre (D).

f (m)

Characteristics of Image formation by a concave mirror:

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Characteristics of Image formation by a convex mirror:

Characteristics of Image formation by a concave lens:

Characteristics of Image formation by a convex lens:

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Image Formation by Concave Mirror

Image Formation by Convex Mirror 

When the object is placed at infinity

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In convex mirror if the object is present at infinity then the image is formed behind the mirror at the principal focus, which is highly diminished, virtual and erect. 

When the object is placed anywhere between pole and infinity

Image is formed behind the mirror, between the pole and the principal focus, which is diminished, virtual and erect.

Refraction through glass slab: When a ray of light passes through a rectangular glass slab, ¡t gets bent twice at the air- glass interface and at the glass- air interface. The emergent ray is parallel to the incident ray and ¡s displaced through a distance.

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Image Formation in Convex Lens Using Ray Diagrams

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Image Formation in Concave Lens Using Ray Diagrams

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Key Points and Concepts:  

The ability or the property of the eye lens to adjust its focal length in order to be able to focus both near and distant objects is known as the power of accommodation. , The minimum distance at which objects can be seen most distinctly without strain is called the least distance of distinct vision.

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       

Retina contains light sensitive cells known as rod and cones. These cells get activated upon illumination and generate electrical signals or pulses. The electrical signals are sent to the brain through optic nerves. . In the brain, the signals are processed, interpreted and the objects in front of the eye are perceived. The process of splitting up of white light into its constituent colour as it passes through a refracting medium is known as dispersion of light. The phenomenon of scattering of white light by colloidal particles is known as Tyndall effect. The sky appears blue due to atmospheric refraction and scattering of light. Rainbow is formed due to dispersion and refraction. In the morning, the sunlight covers a larger distance and thick layers of atmosphere, so except the red colour light, most of the other colours are scattered into the atmosphere. As only red colour light reaches our eye, the morning sun appears to be red.

Important Graphs and Diagrams: The Human Eye:

Refraction through a glass prism:

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Myopia and its correction:

Hypermetropia and its correction:

Recombination of the spectrum of white light: Explanation: It is due to different refractive index of different components of white light. White light has seven colours, namely, violet indigo, blue, green, yellow, orange and red (remembered by the word VIBGYOR). In air (strictly in vacuum) light waves of all colours have same velocity (3 × 108 m/s).

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But in a denser medium, their velocities become less and different. Red light waves, being longest in length, travel fastest and have maximum velocity. Violet light waves, being shortest in length, travel slowest and have minimum velocity in the denser medium. The refractive index ( )of a medium for a wave is given by the relation.  =

Velocity of Wave in air (or vacuum ) Velocity of wave in the medium

=

c v

As  is maximum for violet so it bends maximum A

R

V Red

hite light

Violet B

R O Y G B I V

C

Dispersion of white light by a glass prism

When this dispersed white light is made to fall on a white screen, we get a seven coloured band or light. This coloured band is called spectrum.

Rainbow Formation:

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Key Points and Concepts: Advantages of Harvesting:  

It ensures water availability in non-rainy season. Water becomes available for drinking as well as irrigation.

The 3R's: Reduce, Recycle and Reuse.

Some examples of 3R's: 1. 2. 3. 4.

By switching off unnecessary lights and fans to save electricity. By repairing leak taps to save water. Recycling, plastic, paper, glass and metal items to make required things. The used envelopes carfbe reversed and can be used again instead of throwing away.

Need of sustainable management: Our natural resources are limited. With the rapid increase in human population, due to improvement in health care, the demand for all resources is also increasing. Sustainable management is necessary to provide the economic well-being to the present and the future generations and to maintain a healthy environment and life support system.

Components of fossil fuels: (i) Carbon, (ii) Hydrogen, (iii) Nitrogen, (iv) Sulphur.

Reason to reduce use of fossil fuel:  

These are non-renewable and causes pollution. They contribute to global warming.

Advantages of water stored in ground: 1. 2. 3. 4. 5.

Water spreads to recharge wells. Provides moisture to vegetation over a wide area. Does not provide breeding ground to mosquitoes. Protected from contamination. Advantages of Dams : Irrigation, producing electricity:

Disadvantages of building dams: 1. Displace large number of peasants and tribals without adequate compensation or rehabilitation. 2. They swallow up huge amounts of public money.

Forest Stakeholders: 1. The people living in or around the forests who are directly dependent on forest product. 2. The Forest Department of the government which owns the land and controls the resources from forests.

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3. The industrialists who use the forest produce, but are not dependent on the forest of a particular area. 4. The wildlife and nature enthusiasts, who want to conserve nature in its present form.

Khadin System of Water Harvesting:

Coal and Petroleum Coal: Coal is a blackish solid fossil fuel that occurs in seams inside earth. Uses:     

Fuel for domestic and commercial establishments. Production of electricity in thermal plants. Manufacture of coke Gasification and liquefaction yields fuel gas and synthetic petrol. A number of organic compounds can be obtained from coal-benzene, toluene, phenol, anilines, naphthalene.

Petroleum: Petroleum is a dark coloured liquid fuel that is taken out from various depths of earth both on land and sea shore. It is called mineral oil or rock oil. Petroleum extracted from earth is called crude oil. With the help of fractional distillation Uses:     

LPG is liquefied petroleum gas. It is used in cooking and heating. Petrol, diesel and kerosene are used as a fuel for internal combustion engines of transport vehicles. Lubricating oils provide lubrication to machinery. Paraffin wax is used in candles, Vaseline and water proofing Asphalt is source of several chemicals and dye. It is also used in carpeting of road.

Pollution: Fossil fuels are formed of carbon, hydrogen, nitrogen and sulphur. On burning they produce carbon dioxide, water, oxides of sulphur and nitrogen. In the presence of insufficient oxygen, carbon monoxide is produced instead of carbon dioxide. Carbon Monoxide: It readily combines with haemoglobin forming carboxyhaemoglobin. The latter is unable to combine with oxygen. Therefore, carrying capacity of blood is reduced. In

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closed rooms, it may cause death due to asphyxiation. This is common in winter when coal fired heating device is used. Carbon dioxide: Being a green house gas, it adds to global warming. Nitrogen Oxides: They cause necrosis and killing of plant parts. Internal injuries, eye irritation and loss of smell are produced in human and animals. Corrosion occurs in metals. They are also component of acid rain. Sulphur Oxides: They kill lichens, damage metals, marble and other articles, cause eye irritation, damage to respiratory tract and cause acid rain. Fly ash: Burning of coal also produces particulate matter called fly ash it contains toxic ingredients. Management of Fossil Fuels: The management of fossil fuels is mainly based on better use through use of more and more efficient machines. The internal combustion engines used by vehicles employed in transportation are concentrating on ensuring complete combustion. It will reduce air pollution and increase efficiency. 5-10% ethanol (ethyl alcohol) is being added to petrol to reduce consumption of the latter. Hybrid engines using hydrogen and gasoline are also being developed.

Key Points and Concepts: How non-biodegradable substance affect:  

They produce foul smell during decomposition process. They may produce harmful gases such as ammonia, methane and carbon dioxide. .

Some eco-friendly activities:    

Gardening and planting trees. Use of gunny bags / paper bags in place of polythene. Use of compost and vermi-compost in place of fertilizers. Separation of biodegradable and non-biodegradable substances.

10% Law: There is only 10% flow of energy from one trophic level to the next higher level. Due to this energy loss, only4 or 5 trophic levels are present in each chain. It is known as Ten Percent law. Example: Grass

—>

(4000 J)

Grasshopper —>

Frogs

(400 J)

(40 J)

—>

Snakes (4 J)

Significance of food chain: 

It helps in understanding the food relationship and interactions among various organisms in an ecosystem.

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 

It helps in following the basic mechanism of transfer of food energy and nutrients through various components of nature. It helps to understand the movement of toxic substances in an ecosystem and the problem of their biological magnification.

Ozone Depletion: The amount of ozone in ozonosphere began to drop in 1980s. A spring time ozone hole (area of extreme thinness of ozone) was discovered over Antarctica in 1985 by Farman et al. Its area has been spreading subsequently. An ozone reduction of 1% increases the amount of high energy UV –B radiations reaching the earth by 2%. It means that there has been increase in UV –B radiations reaching the earth by 15 –20%. ODS: Ozone depleting substances or ODS are those substances which react with ozone present in the stratosphere and destroy the same. The main ozone depleting substances are chlorofluorocarbons (CFCs), halons, nitrous oxide, methane, carbon tetrachloride and chlorine. Chlorofluorocarbons are highly stable, odourless, synthetic gaseous substance used as aerosol propellants, coolants, refrigerants, blowing agents, etc. Effects of Ozone Depletion: 1. Cancers – The incidence of skin cancer and herpes would increases. 2. Eye Sight – There will be dimming of eye sight, photoburning and higher incidence of cataracts. 3. Immune system – It will be impaired resulting in increasing number of diseases. 4. Mutations – More mutations will occur. Most of them would be harmful. 5. Photosynthesis – 10–25% decline in photosynthesis would occur. 6. Global warming – Reduced photosynthesis will increase CO2 concentration causing global warming. 7. Damage to Articles

Sample Food Web:

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Sample Food Chains:



Ozone (O3) is a molecule formed by three atoms of oxygen. U-V O2    O+O

O+O 



 O3

Due to ozone layer depletion, the ultraviolet rays reaches the earth and causes certain illeffects which are bad for us and for crops. Exposure to U-V rays can lead to greater incidence of skin cancers, cataracts and damages the eye and immune deficiency. Four appliances which releases Chlorofluorocarbon: (a) Aerosol spray (b) Air conditioner (c) Refrigerator (d) Coolant

Trophic Levels:

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Flow of energy in an ecosystem:

TERM-2

Fundamentals:     

2 General form of quadratic equation is : ax  bx  c = 0, Where a ≠ 0. 2

a is coefficient of x , h is coefficient of x , c is called constant term. Equation of the form ax2 + c = O is called pure quadratic equation. The value of variable satisfying equation is called root. Quadratic equation has two roots.

In order to solve the given Quadratic Equation: 1. Clear the fractions and brackets, if given. 2. By transfering each term to the left hand side; express the given equation as; ax2 + bx + c = 0 or a + bx + cx 2 = 0 3. Factorise left hand side of the equation obtained (the right hand side being zero). 4. By putting each factor equal to zero; solve it.

Methods of Solving Quadratic Equation: 

Factorisation Method (splitting the middle term)

 

Completing the Square Method Quadratic Formula Method (Sridharacharya Formula)

Factorisation Method: Step 1: Resolve the equation in factor using splitting the middle term methods,

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i.e. ax2  bx  c   Ax  B  Cx  D Step 2: Put both factors equal to zero, Ax + B = O and Cx + D = O

Therefore x 

B A

and x



D C

are two roots

Completing the Square Method: Step 1: Let the equation be: ax

2

 bx  c = O

Step 2: Make the coefficient of x2 unity by dividing whole equation by it, you will get: x 2 

b a

c

x

a

0

Step 3: Shift the constant term on RHS, you will get: x2 

b a

x

c a

Step 4: Add the square of half of the coefficient of x. On the both sides of the equation: 2

c b  b b x  2   x         a  2a   2 a   2a 

2

2

Step 5: Write LHS as the perfect square and simplify RHS: 2

b  b 2  4ac  x      2a  2a  Step 6: Take square root on both sides: x  

b

2a



b2  4ac

2a

Step 7: Evaluate the values of x by shifting the constant term on RHS. x 

b  b2  4ac 2a

Quadratic Formula Method: Direct formula to calculate the roots is given as: x 

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b  b2  4ac 2a

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Proof: Use steps given in completing the square method. Discriminant: It is denoted by ‘D’ and given by: D = b2  4ac

Nature of Roots of Quadratic equation: The nature of the roots depends on the value of b 2 – 4ac. b2 – 4ac is called the discriminant of the quadratic equation ax2 + bx + c = 0 and is generally, denoted by D.  D = b2 – 4ac

If D > 0, i.e., b2 – 4ac > 0, i.e., b 2 – 4ac is positive; the roots are real and unequal. Also, (i) If b2 – 4ac is a perfect square, the roots are rational and unequal. (ii) If b2 – 4ac is positive but not perfect square, the roots are irrational and unequal. If D = 0, i.e., b2 – 4ac = 0; the roots are real and equal. If D < 0, i.e., b2 – 4ac < 0; i.e., b 2 – 4ac is negative; the roots are not real, i.e., the roots are imaginary .

Arithmetic Progression is defined as a series in which difference between any two consecutive terms is constant throughout the series. This constant difference is called common difference. Sequence: An arrangement of numbers which has a pattern, which can suggest the successor of every number in the arrangement. e.g.:

1. Arithmetic Progressions 3, 5, 7,9,11 …. —8,—5,—2, 1, 4, 7 ……. 6, 1, —4,—9,—14 ……… 2, 8, 18, 32, 2, 2 2,3 2, 4 2,..........Yes it is Al because it can be written as below: 3  2,3  2 2, 3  3 2,3  4 2,...........

2. Geometric Progressions (not in course) 3, 9, 27, 81 …… 6, 18, —54, 162, —486 ……..

2, 2, 2 2, 4, 4 2, 2, 8......... An Arithmetic Progression (AP): is a list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

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This fixed number is called common difference, denoted by 1d’. It can be +ve, — ve or zero. So general form of an AP is given by: a, a + d, a +2d, a + 3d, a + 4d,+ …………, a + (n -1) d. Where a = first term, d = common difference and a + (n-1) d = n th term.

 d t3 - t2  d t4 - t3  d so on..........

Here

t2 - t1

Hence It can be written:

 t3 - t2 2t2  t1  t 3

t2 - t1

Conclusion: If three numbers a, h and c are in AP then: 2b = a + c. th

 n  1  (i) If no. of terms in any series is odd then only one middle term is exist which is   term  2  where n is odd.

(ii) If no. of terms in any series is even then middle terms are two which are given by th

(n/2) th and

 n      1 term where n is even.  2  

Some Standard Results: (i) Sum of first n natural numbers n



 r  n(n2 1) r  1

(ii) Sum of first n odd natural numbers n



 (2r 1) = n

2

r  1

(iii) Sum of first n even natural numbers n

 2r = n ( n+ 1) r  1

(iv) Sum of squares of first n natural numbers n

 r  n(n 1)6(2n  1) 2

r 1

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(v) Sum of cubes of first n natural numbers n

 n (n  1)  r     2  r  1



2

3

(vi) If for an A.P. p th term is q, q th term is p then m th term is = p + q – m (vii) If for an AP sum of p terms is q, sum of q terms is p, then sum of (p +q)term is: (p+q). (viii) If for an A.P. sum of p terms is equal to sum of q terms then sum of (p + q) terms is zero.

Important Formulae: nth term of an AP

T n = a + (n — 1) d

Sum of the n terms of an AP is: Sn Also: Sn

n

 [2a  (n  1)d ] 2

n

 [a  l ] 2

Where, l = last term, i.e. l = a  (n 1)d

Tips: 1. To an AP if we (i) add (ii) subtract (iii) multiply or (iv) divide each term by the same number, the resulting sequence would always be an AP 2. If three numbers a, b and c are in AP then: 2b = a + c 3. Whenever you be asked to take three numbers which are in AI always take: a - d, a, a + d.

Proof of “Sum of n terms of an AP” We know that general form of an AP is given by: a, a  d , a  2d , a  3d , a  4d ,............, a  (n  a)d

..... (i)

 Sn  a  (a  d )  (a  2d )  (a  3d )  ..........  [a  (n  2)d ]  [a  (n  1)d ]

.... (ii)

Now write the above equation in reverse order,

 Sn  [a  (n 1)d ]  [a  (n  2)d ]  ........  (a  3d )  (a  2d )  (a  d )  a

.... (iii)

Adding the corresponding terms of eq (i) & (ii), we get

 2Sn  [2a  (n  1)d ]  [2a  (n  1)d ]  [2a  (n  1)d ]  .......  [2a  (n 1)d ]  2Sn  n[2a  (n  1)d ] n

 Sn  [2a  (n  1)d ] 2

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n

 [2a  (n 1)d ]

Also, it can be written as: Sn

Sn

2

n

 [a  l ] 2

(where last term, l = a  (n  1)d )

Hence Proved.

Some important key points: 1. We know an

 a   n 1 d

an

 a  nd  d

an

 a  d  nd

i.e. linear equation denotes general term where (i) coefficient of n is common difference‘d’ (ii) Constant term is (a — d).

Fundamentals: 1. Consider a circle C (O, r) and a line PQ. There can be three possibilities given below:   

2. 3. 4. 5.

Non intersecting line w.r.t circle A line intersects circle in two distinct points, this Line is called a Secant. A line which intersects circle exactly at one point is called a Tangent.

From a point P inside a circle, the number of tangents drawn to the circle = O. From a point P on a circle, the number of tangents drawn to the circle = 1. From a point P outside the circle, the number of tangents drawn to the circle = 2. The distance between two parallel tangents drawn is equal to the diameter of the circle.

Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact. Given: A circle with centre O and a tangent AB at a point P of the circle. To prove: OP  AB.

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O A

P

R Q B

Construction: Take a point Q, other than P, on AB. Join OQ. Proof : Q is a point on the tangent AB, other than the point of contact P. 

Q lies outside the circle.

Let OQ intersect the circle at R. Then, OR < OQ [a part is less than the whole]

...(i)

But, OP = OR [radii of the same circle].

....(ii)



OP < OQ [from (i) and (ii)].

Thus, OP is shorter than, any other line segment joining O to any point of AB, other than P. In other words, OP is the shortest distance between the point O and the line AB. But, the shortest distance between a point and a line is t he perpendicular distance. 

OP  AB.

Theorem 2(Converse of Theorem 1): A line drawn through the end of a radius and perpendicular to it is a tangent to the circle. Given : A circle with centre O in which OP is a radius and AB is a line through P such that OP  AB. O A

P

Q B

To prove: AB is a tangent to the circle at the point P. Construction: Take a point Q, different from P, on AB. Join OQ. Proof: We know that the perpendicular distance from a point to a line is the shortest distance between them. 

OP  AB  OP is the shortest distance from O to AB.



OP < OQ.



Q lies outside the circle

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[ OP is the radius and OP < OQ]. Thus, every point on AB, other than P, lies outside the circle. 

AB meets the circle at the point P only.

Hence, AB is the tangent to the circle at the point P. Theorem 3: The tangent at any point of a circl e is perpendicular to the radius through the point of contact. Theorem 4: The lengths of two tangents from an exter nal point to a circle are equal. Given: A circle C (O, r) and two tangents say PQ and PR from an external point P. To Prove: PQ= PR.

Theorem 4: If two tangents are drawn from an external point then (i)

They subtend equal angles at the centre, and

(ii)

They are equally inclined to the line segment joining the centre to that point.

Given: A circle with centre O and a point A outside it. Also, AP and AQ are the two tangents to the circle. P A

O Q

To prove:  AOP = AOQ and OAP = OAQ. Proof: In  AOP and  AOQ, we have NCERT Solutions

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AP = AQ [tangents from an external point are equal] OP = OQ [radii of the same circle] OA = OA [common] 

 AOP  AOQ [by SSS–congruence].

Hence,  AOP =  AOQ and OAP = OAQ.

To divide a line segment internally in a given ratio m : n, where both in and n are positive integers. Steps of Construction: Step 1. Draw a line segment AB of given length by using a ruler. Step 2. Draw any ray AX making an acute angle with AB. Step 3. Along AX mark off (m + n) points A1 ,A 2 ,.........A m ,A m+1 , …… A m+n , such that

A A1 = A1 A 2 = A m+n-1 A m+n . Step 4. Join B A m+n . Step 5. Through the point A m , draw a line parallel to A m+n B by making an angle equal to A A m+n

B at A m . Suppose this line meets AB at point P. The point P so obtained is the required point which divides AB internally in the ratio m : n.

Alternative Method:

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Y B2 B1 C

A A1

A2

B

A3

X

Steps of Construction: 1. Draw any ray AX making an acute angle with AB. 2. Draw a ray BY parallel to AX by making  ABY equal to BAX. 3. Locate the points A 1, A 2, A 3 (m = 3) on AX and B 1, B2 (n = 2) on BY such that AA 1 = A 1 A 2 = A 2 A 3 = BB1 = B1B2. 4. Join A 3B2. Let it in intersect AB at a point C (see figure) Then AC : CB = 3 : 2 Why does this method work ? Let us see. Here  AA 3C is similar to  AB2C. (Why ?) Then

AA3 BB2

Since by construction, therefore,



AC BC

AA3 BB2

AC BC



3

 , 2

3 2

In fact, the methods given above work for dividing the line segment in any ratio. We now use the idea of the construction above for constructing a triangle similar to a given triangle whose sides are in a given ratio with the corresponding sides of the given triangle. Construction of triangles similar to a given triangle: (a) Steps of Construction: when m < n, Step 1. Construct the given triangle ABC by using the given data. Step 2. Take any one of the three sides of the given triangle as base. Let AB be the base of the

given triangle. Step 3. At one end, say A, of base AB, Construct an acute < BAX below the base AB.

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(b) Steps of Construction: when m > n, Step 1. Construct the given triangle by using the given data. Step 2. Take any one of the three sides of the given triangle and consider it as the base.

Let AB be the base of the given triangle. Step 3. At one end, say A, of base AB, construct an acute angle

 BAX below base AB

i.e., on the opposite side of the vertex C.

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To draw the tangent to a circle at a given point on it, when the centre of t he circle is known. Given: A circle with center O and a point P on it. Required: To draw the tangent to the circle at P. Steps of construction:

(i) Join OP. (ii) Draw a line AB perpendicular to OP at the point P, APB is the required tangent at P,

To draw the tangent to a circle from a point o utside it (external point) when its center is known. Given: A circle with center O and a point P outside it. Required: To construct the tangents to the circle from P. Steps of construction:

(j) Join OP and bisect it. Let M be the mid point of OP. (ii) Taking M as centre and MO as radius, draw a circle to intersect C (O, r) in two points, say A and B, (iii) Join PA and PB. These are the required tangents from P to C (O, r).

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Fundamentals: Circle is defined set of all those points which are at constant distance from fixed point. The fixed point is called centre. The constant distance is called radius. The longest chord passing through centre and whose end point lies on circle is called diameter. Circles with same centers are called Concentric Circles. Circumference: In simple words we can say perimeter of circle is called circumference. But

actually π is defined as the ratio of circumference and diameter of circle. 



Circumference Diameter

Therefore, Circumference = π x Diameter If r is the radius of a circle, then (i) Circumference = 2r or d, where d = 2r is the diameter of the circle. (ii) Area = r2 or d2/4 (iii) Area of semi-circle =

r 2 2

(iv) Area of a quadrant of a circle =

r 2 4

Area enclosed by two concentric circles: If R and r are radii of two con-centric circles, then area enclosed by the two circles = R 2 – r2 =  (R 2 – r2) =  (R + r) (R – r)

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r

R

Some useful results: (i) If two circles touch internally, then the distance between their centres is equal to the difference of their radii. (ii) If two circles touch externally, then the distance between their centres is equal to the sum of their radii. (iii) Distance moved by a rotating wheel in one revolution is equal to the circumference of the wheel. (iv) The number of revolutions completed by a rotating wheel in one minute =

Dist ance movedin on eminute Circumference

Arc, Chord, Segment, Sector of a Circle

Arc: Any portion of circumference. e.g.: APB is minor arc while AQB is major arc. Chord: The line joining any two points on the circle. e.g.: AB. Segment: In figure chord AB divides the circle in two segments i.e., APBA (minor segment) and

AQBA (major segment). Sector: The region bounded by the two radii AO and BO and arc AB is called sector of the circle.

Let

 AOB = θ, where θ is called central angle.

Length of Arc: When sector angle

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 AOB = θ.

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Perimeter of the sector = 2r +

 36 0

× 2r

Perimeter of Segment (shaded) of A Circle: AB + arc ( APB)

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Fundamentals:

1. 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 = a 3 cubic units (iii) Diagonal of the cube =

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3 a units.

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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 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 = (R 2 – 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 (R 2 – 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 3

r2h

6. For a sphere of radius r, we have (i) Surface area = 4r2 (ii) Volume =

4 3

r3

7. If h is the height,  the slant height and r 1 and r2 the radii of the circular bases of a frustum of a cone then 

(i) Volume of the frustum = (r12 + r1r2 + r22) h 3

(ii) Lateral surface area =  (r1 + r2)  (iii)Total surface area = {(r1 + r2)  + r12 + r22} (iv) Slant height of the frustum =

h 2  (r 1  r 2 ) 2

(v) Height of the cone of which the frustum is a part =

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hr 1 r 1  r 2

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(vi) Slant height of the cone of which the frustum is a part = (vii) Volume of the frustum =

h 3

A  A 1

2

r 1

r 1  r 2



 A1A 2 , where A 1 and A 2 denote the areas of

circular bases of the frustum.

Tips:

Angle of Elevation: The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the observer. The angle of elevation of the point viewed is the angle formed by the line of sight with the horizontal.

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When the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object is also shown in diagram that we have to assume a horizontal level at our eyes.

Angle of Depression: The angle of depression of a point on the object being viewed is the angle formed by the line of sight with the horizontal. When the point is below the horizontal level, i.e. the case when we lower our head to look at the point being viewed.

Trigonometric Ratios of Some Specific Angles 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°

Fundamentals: 1. 2. 3. 4. 5.

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

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6. Equation of y-axis is x = 0 7. Point on y-axis: (0, y) 8. 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 4

part of a plane divided by co-ordinate axes.

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.

I II ( – ,+) (+,+) ' III O IV ( – , – ) (+, – )

Y'

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Quadrant

x-co-ordinate y-co-ordinat 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: x 2  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 mn

,y

my2  ny1 mn

How to remember the section formula?

Corollary: If P ( x, y ) is the midpoint, therefore m : n = 1 : 1 x 

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x2  x1 2

,y 

y2  y1 2

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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 3

,y

y1  y2  y3 3

Area of Triangle: Area of triangle having vertices ( x1 , y1 ), ( x2 , y2 ) and ( x3 , y3 ) is given by:

A 

1 2

 x1 ( y2  y3 )  x1 ( y2  y3 )  x3 ( y1  y2 )

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, y 1), B(x2, y 2), C(x3, y 3) and D(x4, y 4) be the vertices of quadrilateral ABCD then area of the quadrilateral ABCD =

1 2

[x1 y 2 –x2 y 1 + x2 y 3 –x3 y 2 + x3 y 4 – x4 y 3 + x4 y 1 – x1 y 4]

Fundamentals:     

Experiment: An operation which can produce some well defined outcomes. Sample Space: It is the total number of possible outcomes of a random experiment. Event: Any subset of sample space is called event. Elementary Event: Each outcome of any random experiment. Sure Event (Certain event): An event which always occurs whenever the random experiment is performed.

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 

Impossible Event: An event which never occurs whenever the random experiment is performed. Favourable Event: The cases which ensure the occurrence of an event.

Probability: Probability P (E) of an event E is defined as:

P ( E ) 

Number of favourable outcomes

P ( E ) 

Favourable Event

Total Number of outcomes

Sample Space

Note : It is obvious that 0  m  n. If an event A is certain to happen, then m = n thus P (A) = 1. If A is impossible to happen then m = 0 and so P (A) = 0. Hence we conclude that 0  P (A)  1 Further, if A denotes negative of A i.e. event that A doesn’t happen, then for above cases m, n; we shall have P (A ) = 

n m n

 1

m n

= 1– P (A)

P (A) + P ( A ) = 1

Complement Events: An event associated with a random experiment denoted by P (not-E) which happens only when E does not happen is called the complement of event E.

 

P E or P  notE   1  P  E  Tips: 1. Sum of the probabilities of all the elementary events of an experiment is 1. P( E1 )  P (E2 )  P (E3 )  .......  P (E n )  1 2. Probability of Sure Event is 1. 3. Probability of an Impossible Event is 0. 4. Probability of any event lies between 0 and 1 (including 0 and 1) i.e., 0  P  E   1. 5. Playing Cards : (i)

Total: 52 (26 red, 26 black)

(ii)

Four suits: Heart, Diamond, Spade and Club - 13 cards each

(iii) Court Cards: 12 (4 Kings, 4 queens, 4 jacks) (iv) Honour Cards: 16 (4 aces, 4 kings, 4 queens, 4 jacks)

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6. Out of 52 cards 26 are red in colour and 26 are black in colour. 7. In each suit there is an Ace, a King, a Queen, a Jack, 10, 9, 8, 7, 6, 5, 4, 3 and 2. 8. King, Queen and Jack are called face cards.

Algebra:

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Power series with real variables:

Integer series:

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Physical Constants:

List of perfect Square roots: √1 √4 √9 √16 √25 √36 √49 √64 √81 √100

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1 2 3 4 5 6 7 8 9

√121

11

√441

21

√144

12

√484

22

√169

13

√529

23

√196

14

√576

24

√225

15

√625

25

√256

16

√676

26

√289

17

√729

27

√324

18

√784

28

√361

19

√841

29

√400

20

√900

30

10

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√961

31

√1444

38

√2025

45

√1024

32

√1521

39

√2116

46

√1089

33

√1600

40

√2209

47

√1156

34

√1681

41

√2304

48

√1225

35

√1764

42

√2401

49

√1296

36

√1849

43

√2500

50

√1369

37

√1936

44

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Perfect Cube Roots chart:

Surface Area and Volume Formulas:

1


SURFACE AREA (SA) of a:

PERIMETER (P) of a:

VOLUME (V) of a:

Equations:

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Formulae Handbook for CBSE Class 10 Science and Maths

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