Mathematics (Algebra and Geometry)

Introduction

Mathematics is the language of all sciences. The subject at the secondary level has great importance in a progressive country like India as it develops various living skill. It is important to note that the subject itself has a separate identity in the life of a man. The challenges due to the tremendous growth in the population, globalization, pollution, competition between countries, natural disasters emphasise the need to develop the curriculum in Mathematics at the secondary level. Knowledge of the subject and skills acquired while learning Mathematics develop the ability to execute, manage, plan with precision. This could be effectively inculcated at the secondary level and is hence a decisive factor in according Mathematics an important place in the scheme of studies of secondary education in the state of Maharashtra.

Mathematics at the primary stage of education develops the skill of quantification of experiences in learners. It helps to develop decision making which is applicable to real life situations in future. In addition, it helps enormously in the development of other disciplines which involves analysis, reasoning and qualification of ideas.

The study of Mathematics has to be graded at different levels viz. primary, higher primary and secondary, higher secondary levels. At the primary stage the focus is on understanding of various concept and fundamentals and on developing the basic skills of calculation. At the secondary level the emphasis is on the capacity of students to apply mathematics in solving problems. Secondly a systematic study of Mathematics as a discipline has to be started and continued for the use of algebraic methods, arithmetical skills. A study of the different applications of mathematics in various fielda like science, geography, economics, social science etc. gives the student a comprehensive and global prespective.

In the beginning of the 21st century, man has developed new disciplines like Information Technology, Genetic Engineering, Biotechnology, etc. on the basis of Mathematics. Thus a supplementary study of Mathematics is necessary to develop in students the skills essential for these new disciplines. We are confident that enriching these skills at the secondary level, will meet the basic and allied needs and fulfil the requirements in future. So, it has become necessary for every student to study and to develop a modicum of mathematical skills at the secondary level.

Objectives

To enable the students to-

1. Acquire knowledge of mathematical terms, symbols, concepts, principles and processes, and the skill of presentation of proofs.
           
2. Understand the terms, symbols, concepts, principles, processes and proofs.

3. Develop the ability to apply mathematical knowledge to solve problems in real life situations.

4. Develop the analytical, logical thinking and problem solving abilities of students.

5. Develop expertise in basic algebraic and arithmetical skills.

6. Develop skills in drawing geometrical figures, diagrams, graphs, charts, etc.

7. Acquire sufficient mathematical tools and techniques necessary for daily and practical problems in life.
.
8. Identify inter – relationship between different parts of the problems and draw logical conclusions.

9. Consolidate the mathematical knowledge and skills acquired at the secondary level.

10. Develop necessary skills to work with modern technological devices such as calculators and computers.

11. Develop awareness of the need for national unity, national integration, protection of the environment, and the observation of small family norms.
      
12. Develop reverence and respect towards great Mathematicians particularly towards Indian Mathematicians.

Top

Std. IX - Algebra

1 Sets

• Introduction
• Methods of writing sets
• Types of sets
• Subset – Proper , Improper subset
• Super set, Universal set Venn diagrams
• Operations on sets
• Relations between various operations
• Number of elements in the set and related results.

2. Real Numbers

• Revision of natural numbers, integers, rational numbers and irrational numbers
• Existence of irrational numbers and their representations on the number line
• Every real number is represented by a unique point on the number line and conversely, every point on the number line represents a unique real number
• Properties of real numbers
• Definition of n th root of a real number
• Surds - Definition
• Forms of surds
• Operations and Laws of surds
• Rationalization of Surds
• Absolute value of real number
• Euclid’s division Lemma
• Fundamental theorem of Arithmetic

3. Algebraic Expressions

• Introduction to algebraic expression
• Operations on algebraic expressions
• Methods of factorization of algebraic expression
• Introduction to polynomials
• Operations or polynomials
• Value of polynomials
• Zeors / roots of polynomials
• Relation between zeors and coefficient of polynomials
• Remainder theorem
• Factor theorem

4. Graphs

• Cartesian co-ordinate system
• Understanding of graphs of lines parallel to axes
• Graph of line ax +by+C = 0

5. Linear equations in two variables

• System of linear equations
• Solutions of system of liner equations in two variables
• (Algebraic methods)

6. Ratio, proportion and variation

• Introduction to ratio
• Properties to ratio
• Properties of equal ratios
• Theorem on equal ratios
• Percentage as a ratio
• Introduction to proportion
• Introduction to variation
• Revision of concepts based on Direct variation, Inverse variation
• Mixed variation
• Real life problem based on ratio, proportion and variation

7. Statistics

• Collection of data
• Classification and tabulation of data
• Diagramatic representation of data
• Graphical representation of data
• Mean, median, mode of upgrouped data

Top

Std. IX - Gemotery

1. Lines and Angles

• Instroduction to lineBasic terms and definitions related to line
• Introduction to Euclid’s Geometry
• Plane separation axiom
• Introduction to angles in terms of rotation
• Directed angles  Sexageesimal system
• Types of angles – zero angle, straight angle, coterminal angle
• Relation between angles
• Introduction to mathematical proofs
• Parallel lines
• Results on parallel lines and transversal
• Tests of parallel lines
• Results on perpendicular lines
• Distance of a point from a line

2. Triangles

• Types of triangles
• Terms related  to triangle
• Properties of triangle
• Exterior angles and corresponding interior opposite angles
• Results involving exterior angle and corresponding interior opposite angles
• Similar triangles

3. Congruence of triangles

• Criteria of congruent triangles
• Theorem of an isosceles triangle and its converse
• Perpendicular bisector theorem
• Angle bisector theorem
• Properties of triangles based on inequalities
• Property of perpendicular drawn from a point outside the line.

4. Circle

• Introduction to circle and related terms
• Circle passing through the given points
• One and only one circle passes through the three non collinear points
• Congruence of circles
• Properties of chords of the circle

5. Quadrilateral

• Properties of quadrilateral
• Properties of parallelogram
• Properties of rectangle
• Properties of a trapezium
• Properties of a rhombus
• Properties of a square
• Kite
• Tests for particular quadrilateral
• Theorem on midpoints of two sides of a triangle and its converse
  

6. Coordinate Geometry

• Distance formula
• Section formula
• Area of a triangle

7. Geometric constructions

• Basic construction – perpendicular bisector of given segment.
• Construction of a triangle – sum / difference of two sides and base angles is given
• Perimeter and base angles are given

8. Trigonometry

• Introduction to trigonometric ratios
• Trigonometric Ratios of angles 0⁰, 30⁰, 45⁰, 60⁰, 90⁰
• Trigonometric identities for complementary angles

9. Mensuration

• Area of triangle
• Area of regular hexagon, polygon
• Area of quadrilaterals
• If two parallelogram lie between two parallel lines and have the same base then they have the same area.
• If a triangle and parallelogram have the same base and lie between the same parallel lines, then the area of the triangle is half the area of the parallelogram.
• Perimeter of triangle and quadrilateral
• Area of circle

Top

Std. X - Algebra

1. Arithmetic Progression

• Introduction to Sequence
• Arithmetic progression ( A.P.) and Geometric Progression (G.P)
• General terms of an A.P. and G.P.
• Sum of the first ‘n’ terms of an A.P. and G.P.
• Arithmetic Mean and Geometric Mean

2. Quadratic Equations

• Introduction to quadratic equations
• Solutions of quadratic equations
• Nature of roots based on discriminant
• Relation between roots of the equation and coefficient of the equation
• Equations reducible to quadratic form

3. Linear equations in two variables

• System of linear equations in two variables
• Algebraic methods of solving linear equations in two variables
• Graphical representation of different possibilities of solutions / inconsistency
• Graphical method of solving a system of linear equations
• Determinant of order two
• Cramer rule
• Consistency of pair of linear equations

4. Probability

• Introduction to probability and related terms
• Classical definition of probability
• Types of events
• Equally likely outcomes
• Probability of an event
• Properties of Probability
• Addition theorem ( without proof)

5. Statistics

• Brief revision of Tabulation of data, inclusive and exclusive type of tables
• Mean median and mode of grouped data
• Histograms, frequency polygon, frequency curve, pie digram
• Ogives ( Cumulative frequency graphs)
• Applications ogives in determination of median
• Relation between measures of central tendency
• Introduction to normal distribution
• Properties of normal distribution

Top

Std. X - Geometry

1. Similarity

• Properties of ratios of area of two triangles
• Basic proportionality theorem
• Introduction to similarity
• Similar triangles
• Areas of two similar triangles
• Similarity in right angled triangles
• Pythagoras theorem and its converse
• 30⁰ – 60⁰ – 90⁰ theorem and 45⁰ – 45⁰ – 90⁰ theorem
• Application of Pythagoras theorem in acute and obtuse angle.
• Appolonius theorem

2. Circle

• Tangents and its properties
• Theorem – Tangent at any point to the circle is perpendicular to the radius and it converse
• Number of tangents from a point to a circle
• Theorem – The length of two tangent segments drawn from a point outside the circle are equal
• Touching circles
• Introduction to an arc
• Angle subtended by the are to the centre and to the point on the circle
• Cyclic quadrilateral
• Tangent – Secant theorem

3. Co-ordiante Geometry

• Slope of a line
• Intercepts made by a line
• Standard forms of equation of a line
• General equation of a line

4. Geometric Constructions

• Division of line segment in a given ratio
• Basic geometric constructions
• Construction of tangent to the circle from the point on the circle and outside the circle.
• Construction of tangent without using centre
• Construction of triangle 0 If the base, angle apposite to it and either median or altitude is given
• Construction of a triangle similar to given triangle

5. Trigonometry

• Angles in standard position
• Trigonometric ratios in terms of coordinates of point
• Trigonometric Identities ( with proof) Sin (0) = -Sin 0 ; Cos (-0) = Cos 0
• Use of basic identities and their applications
• Problems on height and distance

6. Mensuration

• Length of an area
• Area of the sector
• Area of a Circular Segment
• Euler’s formula
• Surface area and volume of cuboids Spheres, hemispheres, right circular cylinders, cones, frustum of a cone.
• Problems based on area and perimeter / circumference of circle, sector and segment of a circle.
• Problems on finding surface area and volumes of combinations of any two of the following : cuboids, spheres, hemispheres and right circular cylinders/ cones.
• Problems involving converting one type of metallic solid and into another

Top

Science And Technology

The National curriculum Framework 2005 (NCF 2005) is one of the most comprehensive documents published in the last decade. It suggests radical changes in curricula for the country and offers an excellent framework for preparing need based curricula. While revising the syllabi for Science and Technology, the position paper on science ( NCF 2005) has selected ‘’Learning without burden” as the main theme. It also recommends a pedagogy which is hand-on and inquiry based.
The present syllabus of Science and Technology for STD IX and X is based on the principles and themes suggested in NCF 2005. The Themes are cross – disciplinary in nature : Food, Materials, The world of the Living, How things work, Moving things; people and ideas; Natural phenomena and Natural Resources. Some themes have been merged to consolidate content.

As suggested in NCF 2005, unnecessary focus on enumeration has been avoided. More importance has been given to the processes in science rather than focusing on only factual information.

Since the themes are inter-linked to each other, the entire syllabus has been integrated into one paper. This will facilitate better understanding of the subject across disciplinary boundaries and at the same time expose students to many topics in Biology, physics and Chemistry.

The Objectives are :

1. To enable the students to ‘’Learn without Burden”.
2. To expose the students to a “hand-on way of learning science.
3. To correlate scientific principles to the student’s experience.
4. To involve students in exploring topics through discussion and activity.
5. To imbibe the spirit of enquiry in students through valuable learning experiences through experiment.
6. To help the students become autonomous learners.

Top

Std. IX - Science and Technology

1. Food

• Plant and animal breeding and selection for quality improvement.
• Use of fertilizers, manures
•  Protection from pests and diseases; organic farming.

2. Materials

• Matter : All thing occupy space, possess mass. Definition of matter, characteristics of solids, liquids and gases e.g. shape, volume, density. Change of state : freezing, melting, evaporation, condensation, sublimation. Cooling by evaporation. Absorption of heat.
• Elements, mixtures and Compounds : elements, compounds and mixtures as types of chemical substances. Types of mixtures; Heterogeneous, homogeneous, colloids, suspensions.
• Combination of substances : Law of constant proportion, atomic and molecular masses. Particle nature, basic units : Atoms and Molecules.
•  Mole Concept : Relationship of mole to mass of the particles and numbers. Valency. Chemical formulae of common compounds.
• Atomic Structure : Electrons, Protons and neutrons. Atomic number and atomic mass number. Isotopes and Isobars.

3. The World of the Living

• Biological Diversity
  • Diversity  of plants and animals : Basic issues in scientific naming. Basis of classification, Hierarchy of categories/ groups.
  • Plant classification : Major Plant groups ( salient features) : Bacteria, Thallophyta, Bryophyta, Pteriodphyta, gymnosperms and Angiosperms.
  • Animal Classification : Major groups of animals ( salient features) : Non-Chordates up to phyla and non-chordates up to classes.
• Smallest living unit of Life
  • Cell : Cell as a basic unit of life prokaryotic and eukaryotic cells, multicellular organisms; cell membrane and cell wall, cell organelles, chloroplast, mitochondira, vacuoles, ER, Golgi apparatus. Nucleus, chromosomes, basic structure, number.
  • Tissues, organs, organ system, organism
  • Structure and functions of animal and plant tissues. ( four types in animals; meristematic and permanent tissues in plants).
• Health
  • Failure of health leading to disease.
  • Disease and its causes.
  • Diseases caused by microbes and their prevention : Typhoid, diarrheoa, malaria, hepatitis, rabies, AIDS, TB , polio.
  • Pulse Polio program
• Exchange of substances by living organisms with the external world
  • Diffusion / exchange of substances between cells and their environment and between the cell themselves in the living system; role in nutrition, water and food transport, excretion, gaseous exchange.

4. Moving Things. People and ideas. Motion

• Motion : displacement, velocity, Uniform and non-uniform motion along a straight line, acceleration, distance-time and velocity time graphs for motion and uniformly accelerated motion. Equations of motion by graphical method. Elementary idea of uniform circular motion.
• Force and Newton’s Laws : Force and Motion; Newton’s Laws of Motion. Interia of a body,interia and mass, momentum , force and acceleration
• Elementary idea of conservation of momentum, action and reaction forces.
• Graviatation :  gravitation, universal law of gravitation, force of gravitation of the –earth  (earth’s gravity), acceleration due to gravity, mass and weight, free fall.
• Work, Energy and Power : work done by a force, energy, power. Kinetic and potential energy, Law of conservation of energy.
• Floating Bodies : Thrust and Pressure, Archimedes, Principle, Buoyancy. Idea of relative density
• Sound ; Nature of sound and its propagation through different media, speed of sound, range of hearing in humans; ultrasound; reflection of sound; echo and SONAR
• Structure of Human Ear ( Auditory aspect).

1. To prepare

• A true solution of common salt, sugar and alum
• A suspension of soil, chalk powder and fine sand in water
• A colloidal of starch in water and egg albumin in water and distinguish between these on basis of
  • transparency
  • filtration criterion
  • stability

2. To prepare

• A mixture
• A compound

Using iron filings and sulphur powder and distinguish between these on the basis of :

• Apperance i.e. homogeneity and heterogeneity
• Behaviour towards a magnet
• Behaviour towards carbon disulphide as a solvent
• Effect of heat

3. To carry out the following chemical reactions and record observations. Also identify the type of reaction involved in each case

• Iron with copper sulphate solution in water.
• Burning of Magenisum in air.
• Zinc with dilute sulphuric acid
• Heating of Lead Nitrate
• Sodium sulphate with Barium chloride in the form of their solutions in water.

4. To verify laws of reflection of sound.

5. To determine the density of solid. (denser than water) by using a spring balance and a measuring cylinder.

6. To establish the relation between the loss in weight of a solid when fully immersed in

• tap water
• Strongly salty water, with the weight of water displaced by it by taking at least two different solids.

7. To measure the temperature of hot water as it cools and plot a temperatue-time graph.

8. To determine the velocity of a pulse propagated through stretched string/ slinky.

9. To prepare stained temporary mounts of onion peel and record observation and draw labeled digram.

10. To identify parenchyma and sclerenchyma  tissues in plants, striped muscle fibers and nerve cells in animals, from prepared slides and to draw their labelled digrams.

11. To separate the components of a mixture of sand, common salt and ammonium chloride ( or camphor ) by Sublimation.

12. To determine the melting point of ice and the boiling point of water

13. To test (a) the presence of starch in the given food sample (b) presence of the adulterant metanil yellow in dal.

14. To study the characteristic of spirogyra / Agaricus, Moss /Fern, Pinus ( either with male or female cone) and an Angiospermic plant. Draw and give two identifying features of groups they belong to.

15. To observe and draw the given specimens – earthworm, cockroach, bony fish and bird. For each specimen record

• One specific feature of – its phylum
• One adaptive feature with reference to its habitat.

Top

Std. X - Science and Technology

1. Materials

• Acids and Bases : Acids and Bases and Salts : General properties, examples and uses.
• Chemical Reactions : Types of chemical reactions : combination, decomposition, displacement, double decomposition, precipitation, and neutralization.
•  Oxidation and Reduction : Oxidation and Reduction in terms of gain and loss of oxygen and hydrogen.
• Metals and Non-Metals : Brief discussion on basic metallurgical processes. Properties of common metals. Elementary idea about bonding.
• Carbon Compounds : Elementary idea about bonding. Saturated hydrocarbons, alcohols, carboxylic acids ( Properties only)
• Common chemicals used in daily life : Soap, common salt, Washing Soda, Banking Soda, bleaching powder, Plaster of Paris.
• Classification of Elements : Brief historical account, Mendeleev’s periodic table, gradation in properties.

2. The World of the Living

• Life Processes : Definition of “Living things”. Basic concept of nutrition, respiration, transport and excretion in plants and animals.
• Control in the Living : Tropic movements in plants. Introduction to plant hormones; control and coordination in animals; voluntary, involuntary and reflex action, nervous system, chemical coordination, animal hormones.
• Reproduction in the Living : Reproduction in plants and animals. Need for and methods of family planning. Safe Sex vs – HIV/AIDS . Child bearing and women’s health.
• Heredity and Evolution : Heredity, origin of life –brief introduction, Basic concepts of evolution.

3. Moving things, people and Ideas

• Electric Circuits : Potential  and potential difference. Ohm’s Law, Resistances in series and parallel. Power dissipated due to current. Inter relation between P,V, I and R.
• Magnets : Magentic field, field lines. Field due to a current carrying wire, coil, solenoid, Force on current carrying conductor. Fleming’s Left Hand Rule. Electric motor, electromagnetic , induction. Induced potential differences, induced current. Electric  generator, principle and working. Direct and Alternating current. Frequency of AC. Advantages of Ac over DC. Domestic Electric Circuits.

4. Natural phenomena

• Light : Convergence and Divergence of Light.
• Spherical Mirrors : Images formed by a concave mirror. Related concepts : centre of curvature, principle axis, optical centre, focus, focal length.
• Refraction : Refraction ; appreciation of the concept of refraction. Laws of refraction. Velocity of light. Refractive index; twinkling of stars; dispersion of light. Scattering of light
• Lenses : Images formed by a convex lens; functioning of the lens in the human problems of vision and their remedies. Application of spherical mirrors and lenses.

    
Practicals - List of experiments

1. To find the p H of the following samples by using pH paper /universal indicator.

• Dilute Hydrochloric acid
• Dilute NaOH solution
• Dilute Ethanoic and solution
• Lemon juice
• Water
• Dilute Sodium Biacarbonate Solution.

2. To study, the properties of acids and bases HCl & NaOH by their reaction with

• Litmus solution ( Blue/Red)
• Zinc metal
• Solid Sodium Carbonate

3. To determine the focal length of

• Concave mirror
• Convex lens by obtaining the image of a distant object.

4. To trace the path of a ray of light passing through a rectangular glass slab for different angles of incidence.

Measure the angle of incidence, angle of refraction, angle of emergence and interpret the result

5. To study the dependence of current (I) on the potential difference (V) across a resistor and determine in resistance. Also plot a graph between V and I .

6. To determine the equivalent resistance of two resistors when connected in series.

7. To determine the equivalent resistance of two resistors when connected in parallel.

8. To prepare a temporary mount of a leaf peel to show stomata.

9. To show experimentally that light is necessary for photosynthesis

10. To show experimentally that carbon dioxide is given out during respiration.

11. To study (a) binary fission in Amoeba and (b) budding in yeast with the help of prepared slides.

12. To determine the percentage of water absorbed by raisins.

13. To perform and observe the following reactions and classify them into :

• Combination Reaction
• Decomposition  Reaction
• Displacement Reaction
• Double Displacement Reaction
  
  
  • Action of water on quick lime
  • Action of heat on Ferrous Sulphate crystals
  • Iron Nails kept in copper sulphate solution
  • Reaction between Sodium sulphate and Barium chloride solutions.

14. To observe the action of Zn, Fe, Cu and Al metals on the following salt

• ZnSO₄ (aq.)
• FeSO₄ (aq.)
• CuSO₄ (aq.)
• Al₂(SO₄)₃ (aq.)

Arrange Zn, Fe, Cu and Al metals in the decreasing order of reactivity based on the above result.

15. To study the following properties of acetic acid (ethanoic acid) :

• Odour
• solubility in water
• effect on litmus
• reaction with sodium bicarbonate.

Top