BCECE 2024 Syllabus: Your Ultimate Study Guide

PHYSICS

Physical world and measurement Physics : scope and excitement, nature of physical laws; Physics, technology and society. Need for measurement : Units of measurement, systems of units. S.I. units, fundamental and derived units, length, mass and time measurements, accuracy and precision of measuring instruments, errors in measurement, significant figures, regular and irregular errors. Dimensions of physical quantities, dimensional analysis and its applications. Kinematics Frame of reference, Motion in straight line, position time graph, speed and velocity. Uniform and non-uniform motion, average speed and instantaneous velocity. Uniformly accelerated motion, velocity time and position time graphs, relations for uniformly accelerated motion (graphical treatment). Elementary concepts of differentiation and integration for describing motion. Scalar and vector quantities : Position and displacement vectors, general vectors and notation, equality of vectors, multiplication of vectors by a real number, addition and subtraction of vectors, relative velocity. Unit vector, Resolution of a vector in a plane-rectangular components. Motion in a plane, Cases of uniform velocity and uniform acceleration - projectile motion, uniform circular motion. Laws of Motion Intutive concept of force, Inertia, Newton's first law of motion, momentum and Newton's Second law of motion, impulse, Newton's third law of motion, Law of conservation of linear momentum and its applications. Equilibrium of concurrent forces, static and kinetic friction, laws of friction, rolling friction, lubrication, dynamics of uniform circular motion : centripetal force examples of circular motion (vehical on level circular road, vehical on banked road). Work, Energy and Power Scalar product of Vector work done by a constant force and a variable force, kinetic energy, work-energy theorem, power. Notion of potential energy, potential energy of a spring, conservative forces; conservation of mechanical energy (Kinetic and potential energies), non conservative forces, elastic and inelastic collissions in one and two dimension. Motion of System of Particles and Rigid body. Centre of mass of two-particle system, momentum, conservation and centre of mass motion, centre of mass of a rigid body, centre of mass of circular ring, disc, rod and sphere. Vector product of vectors; momentum of a force, torque angular momentum, conservation of angular momentum with some examples. Equilibrium of rigid bodies, rigid body rotation and equations of rotational motion, comparison of linear and rotational motion, moment of inertia, radius of gyration. Values of M.I. for simple gemoetrical objects (no derivation), statement of parallel and perpendicular axes theorems and their applications. Gravitation Keplar's laws of planetary motion, The universal law of gravitation, Acceleration due to gravity and its variation with altitude and depth. Gravitational potential energy, gravitational potential, escape velocity, orbital velocity of satellite, Geo-stationary satellites. Properties of Bulk Matter Elastic behaviour, stress-strain relationship, Hooke's law, Young's modulus, Bulk modulus, Deformation, Shear modulus of rigidity. Pressure due to fluid column, pascal's law and its applications / hydraulic lift and hydraulic brakes). Effect of gravity on fluid pressure. Viscocity, stoke's law terminal velocity, Reynold's number, Streamline and turbulent flow, Bernoulli's theorem and its applications. 32 Surface energy and surface tension, angle of contact, application of surface tension, ideas to drop bubbles and capillary rise. Heat and Thermodynamics Heat, temperature, thermal expansion, specific heat capacity, Calorimetry, change of state, latent heat. Heat transfer - conduction, convection and radiation, thermal conductivity, Newton's law of cooling. Thermal equilibrium and definition of temperature (Zeroth law of thermodynamics). Heat, work and internal energy. First law of thermodynamics. Second law of thermodynamics, Reversible and irreversible processes. Heat engines and refrigerator. Behaviour of Perfect Gas and Kinetic Theory Equation of state of a perfect gas, work done on compressing a gas. Kinetic theory of gases : Assumptions, concept of pressure, Kinetic energy and temperature, rms speed of gas molecules, degrees of freedom, law of equipartition of energy (statement only) and application to specific heat capacities of gases, concept of mean free path, Avogadro's number. Oscillations and Waves. Periodic motion - period, frequency, displacement as a function of time, periodic functions, simple harmonic motion (SHM) and its equation, phase, oscillation of a spring - restoring force and force constant energy in SHM - Kinetic and potential energies, simple pendulum - derivation of expression for its time period) free, forced and damped oscillations (qualitative ideas only), resonance. Wave motion, Longitudinal and transverse waves, speed of wave motion, Displacement relation for progressive waves, principle of superposition of waves, reflection of waves, standing waves in strings and organ pipes, fundamental mode and harmonics, Beats, Doppler effect. * Electrostatics Electric Charge; Conservation of charge, Coulomb's law-force between two point charges, forces between multiple charges; superposition principle and continuous charge distribution. Electric field; electric field due to a point charge, electric field lines; electric dipole, electric field due to a dipole; torque on a dipole in uniform electric field. Electric flux, statement of Gauss's theorem and its applications to find field due to infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell (field inside and outside). Electric potential, potential difference, electric potential due to a point charge, a dipole and system of charges; equipotential surfaces, electrical potential energy of a system of two point charges and of electric dipole in an electrostatic field. Conductors and insulators, free charges and bound charges inside a conductor. Dielectrics and electric polarisation, capacitors and capacitance, combination of capacitors in series and in parallel, capacitance of a parallel plate capacitor with and without dielectric medium between the plates, energy stored in a capacitor. Van de Graaff generator. Current Electricity Electric current, flow of electric charges in a metallic conductor, drift velocity, mobility and their relation with electric current; Ohm's law, electrical resistance, V-I characteristics (linear and non-linear), electrical energy and power, electrical resistivity and conductivity. Carbon resistors, colour code for carbon resistors; series and parallel combinations of resistors; temperature dependence of resistance. Internal resistance of a cell, potential difference and emf of a cell, combination of cells in series and in parallel. Kirchhoff's laws and simple applications. Wheatstone bridge, metre bridge. Potentiometer - principle and its applications to measure potential difference and for comparing emf of two cells; measurement of internal resistance of a cell. Magnetic effects of current & Magnetism Concept of magnetic field, Oersted's experiment. Biot - Savart law and its application to current carrying circular loop. Ampere's law and its applications to infinitely long straight wire, straight and toroidal solenoids. Force on a moving charge in uniform magnetic and electric fields. Cyclotron. Force on a current-carrying conductor in a uniform magnetic field. Force between two parallel current-carrying conductors-definition of ampere. Torque experienced by a current loop in uniform magnetic field, moving coil galvanometer-its current sensitivity and conversion to ammeter and voltmeter. Current loop as a magnetic dipole and its magnetic dipole moment. Magnetic dipole moment of a revolving electron. Magnetic field intensity due to a magnetic dipole (bar magnet) along its axis and perpendicular to its axis. Torque on 33 a magnetic dipole (bar magnet) in an uniform magnetic field; bar magnet as an equivalent solenoid, magnetic field lines; Earth's magnetic field and magnetic elements. Para-, dia- and ferro- magnetic substances, with examples. Electromagnets and factors affecting their strengths. Permanent magnets. Electromagnetic Induction and Alternating currents Electromagnetic induction; Faraday's law, induced emf and current; Lenz's Law, Eddy currents. Self and mutual inductance. Need for displacement current. Alternating currents, peak and rms value of alternating current/ voltage; reactance and impedance; LC oscillations (qualitative treatment only), LCR series circuit, resonance; power in AC circuits, wattless current. AC generator and transformer. Electromagnetic Waves Electromagnetic waves and their characteristics (qualitative ideas only). Transverse nature of electromagnetic waves. Electromagnetic spectrum (radio waves, microwaves, infrared, visible, ultraviolet, X-rays gamma rays) including elementary facts about their uses. Optics Reflection of light, spherical mirrors, mirror formula, Refraction of light, total internal reflection and its applications, optical fibres, refraction at spherical surfaces, lenses, thin lens formula, lens-maker's formula. Magnification, power of a lens, combination of thin lenses in contact, Refraction and dispersion of light through a prism. Scattering of light-blue colour of the sky and reddish appearance of the sun at sunrise and sunset. Optical instruments; Human eye, image formation and accommodation, correction of eye defects (myopia, hypermetropia, presbyopia and astigmatism) using lenses. Microscopes and astronomical telescopes (reflecting and refracting) and their magnifying powers. Wave optics: wave front and Huygens' principle, reflection and refraction of plane wave at a plane surface using wave fronts. Proof of laws of reflection and refraction using Huygens' principle. Interference, Young's double slit experiment and expression for fringe width, coherent sources and sustained interference of light. Diffraction due to a single slit, width of central maximum. Resolving power of microscopes and astronomical telescopes. Polarisation, plane polarised light; Brewster's law, uses of plane polarised light and Polaroids. Dual Nature of Matter and Radiation Photoelectric effect, Hertz and Lenard's observations; Einstein's photoelectric equation-particle nature of light. Matter waves-wave nature of particles, de Broglie relation. Davission-Germer experiment. Atoms and Nuclei Alpha-particle scattering experiment; Rutherford's model of atom; Bohr model, energy levels, hydrogen spectrum. Composition and size of nucleus, atomic masses, isotopes, isobars; isotones, Radioactivity-alpha, beta and gamma particles/rays and their properties; radioactive decay law. Mass-energy relation, mass defect; binding energy per nucleon and its variation with mass number; nuclear fission and fusion. Electronic Devices Semiconductors; semiconductor diode - I-V characterstics in forward and reverse bias, diode as a rectifier; I-V characteristics of LED, photodiode, solar cell, and Zener diode; Zener diode as a voltage regulator. Junction transistor, transistor action, characterstics of a transistor; transistor as an amplifier (common emitter configuration) and oscillator. Logic gates (OR, AND, NOT, NAND and NOR). Transistor as a switch. Communication Systems Elements of a communication system (block diagram onbly); bandwidth of signals (speech, TV and digital data); bandwidth of transmission medium. Propagation of electromagnetic waves in the atmosphere, sky and space wave propagation. Need for modulation. Production and detection of an amplitude-modulated wave.

CHEMISTRY

Some basic concepts of Chemistry : General Introduction : Importance and scope of chemistry. Historical approach to particulate nature of matter, laws of chemical combination, Dalton's atomic theory; concept of elements, atoms and molecules, Atomic and molecular masses. Mole concept and molar mass; percentage composition, empirical and molecular formula; chemical reactions, stoichiometry and calculations based on stoichiometry. Structure of Atom : Discovery of electron, proton and neutron and their characteristics; atomic number, Isotopes & Isobars, Thomson's model and its limitation, Rutherford's model and its limitations, Bohr's model and its limitations, concept of shells and subshells, dual nature of matter and light, De Broglic's relationship, Heisenberg uncertainty principle, concept of orbitals, Quantum numbers, shapes of S.P. and D orbitals, rules, for filling electrons in orbitals Aufbau principle, Pauli exclusion principle and Hund's rule, electronic configuration of atoms, stability of half filled and completely filled orbitals. Classification of Elements and Periodicity in Properties: Significance of classification, brief history of the development of periodic table, modern periodic law and the present form of periodic table, periodic trends in properties of elements - atomic radii, ionic radii, ionization enthalpy, electron gain enthalpy, electro negativity, valence. Chemical Bonding and Molecular Structure : Valence electrons, ionic bond, covalent bond, bond parameters, Lewis structure, polar character of covalent bond, Covalent characters of Ionic bond, valence bond theory, resonance, geometry of covalent molecules, VSEPR theory, concept of hybridization involving s, p and d orbitals and shapes of some simple molecules, molecular orbital, theory of homonuclear diatomic molecules (qualitative idea only). Hydrogen bonding. States of Matter : gases and liquids : Three states of matter, Intermolecular interactions, type of bonding, melting and boiling points. Role of gas laws in elucidating the concept of the molecule, Boyle's law, Charle's law, Gay Lussac's law Avogadro's law, Ideal behaviour, empirical derivation of gas equation. Avogadro's number. Ideal gas equation. Derivation from ideal behaviour, liquification of gases, critical temperature. Liquid State - Vapour pressure, viscosity and surface tension (qualitative idea only, no mathematical derivations). Thermodynamics : Concepts of system, types of systems, surroundings, work, heat, energy, extensive and intensive properties, state functions, First law of thermodynamics - internal energy and enthalpy, heat capacity and specific heat, measurement of ?U and ?H, Hess's law of constant heat summation enthalpy of : bond dissociation, combustion, formation, atomization, Sublimation, phase transformation, ionization and solution. Introduction of entropy as a state function, free energy change for spontaneous and non-sponteneous process, criteria for equilibrium. Equilibrium : Equilibrium in physical and chemical processes dynamic nature of equilibrium, law of mass action, equilibrium constant, factors affecting equilibrium - Le Chatelier's principle; ionic equilibrium - ionization of acids and bases, strong and weak electrolytes, degree of ionization, concept of pH. Hydrolysis of salts (elementary idea), buffer solutions, solubility product, common ion effect (with illustrative examples). Redox Reactions : Concept of oxidation and reduction, redox reactions, oxidation number, balancing redox reactions, applications of redox reactions. Hydrogen : Position of hydrogen in periodic table, occurrence, isotopes, preparation, properties and uses of hydrogen; hydrides - ionic, covalent and interstitial; physical and chemical properties of water, heavy water; hydrogen peroxide - preparation, reactions and structure; hydrogen as a fuel. s-Block Elements (Alkali and Alkaline earth metals): Group 1 and Group 2 elements : General introduction, electronic configuration, occurrence, anomalous properties of the first element of each group, diagonal relationship, trends in the variation of properties (such as ionization enthalpy, atomic and ionic radii) trends in chemical reactivity with oxygen, water, hydrogen and halogens; uses. Preparation and properties of some important compounds : Sodium carbonate, sodium chloride, sodium hydroxide and sodium hydrogen carbonate, biological importance of 35 sodium and potassium. CaO, CaCO3 and industrial use, lime and limestone. Biological importance of Mg and Ca. Some p-Block Elements General Introduction p-Block Elements Group 3 elements : General introduction, electronic configuration, occurrence, Variation of properties, oxidation states, trends in chemical reactivity, anomalous properties of first element of the group; Boron-physical and chemical properties, some important compounds : borax, boric acids, boron hydrides. Aluminium : uses, reactions with acids and alkalies. Group 4 elements : General introduction, electronic configuration, occurrence, variation of properties, oxidation states, trends in chemical reactivity, anomalous behaviour of first element, Carbon - catenation, allotropic forms, physical and chemical properties; uses of some important compounds : oxides. Important compounds of Silicon and a few uses: Silicon tetrachloride, silicons, silicates and zeolite. (Part-I) Organic Chemistry - Some Basic Principles and Techniques : General introduction, methods of qualitative and quantitative analysis, classification and IUPAC nomenclature of organic compounds. Electronic displacements in a covalent bond; inductive effect, electromeric effect, resonance and hyper conjugation. Hemolytic and heterolytic fission of a covalent bond: free radicals, carbocations, carboanions; electrophiles and nucleophiles, types of organic reactions. Hydrocarbons : Classification of hydrocarbons : Alkanes - Nomenclature, isomerism, conformations (ethane only), methods of preparation, physical properties, chemical reactions including halogenation, free radical mechanism, combustion and pyrolysis. Alkenes - Nomenclature, structure of double bond (ethane), geometrical isomerism, methods of preparation, physical properties, chemical reaction : addition of hydrogen, halogen, water, hydrogen halides (Markovnikov's addition and peroxide effect), ozonolysis, oxidation, mechanism of electrophyllic addition. Alkynes - Nomenclature, structure of triple bond (ethyne), methods of preparation, physical properties, chemical reactions; acidic character of alkynes, addition reactions of hydrogen, halogenes, hydrogen halides and water. Aromatic hydrocarbons - Introduction, IUPAC nomenclature, Benzene : resonance, aromaticity : methods of preparation, chemical properties. Mechanism of electrophilec substitution - nitration, sulphonation, halogenation, Friedal Craft's alkylation and acylation; directive influence of functional group in mono-substituted benzene; carcinogenicity and toxicity. Environmental Chemistry : Environmental pollution - Air, water and soil pollution, chemical reactions in atmosphere smogs, major atmospheric pollutants : acid rain, ozone and its reactions, effects of depletion of ozone layer, greenhouse effect and global warming - pollution due to industrial wastes, green chemistry as an alternative tool for reducing pollution, strategy for control of environmental pollution. * Solid State Classification of solids based on different binding forces : molecular, ionic covalent and metallic solids, amorphous and crystalline solids (elementary idea), unit cell in two dimensional and three dimensional lattices, calculation of density of unit cell, packing in solids, voids, number of atoms per unit cell in a cubic unit cell, point defects, electrical and magnetic properties. Solutions Types of solutions, expression of concentration of solutions of solids in liquids, solubility of gases in liquids, solid solutions, colligative properties--relative lowering of vapour pressure, elevation of Boiling Point, depression of freezing point, osmotic pressure, determination of molecular masses using colligative properties, abnormal molecular mass. Electrochemistry Redox reactions, conductance in electrolytic solutions, specific and molar conductivity variations of conductivity with concentration, Kohlrausch's law, electrolysis and laws of electrolysis (elementary idea), dry cell-electrolytic cells and Galvanic cells; lead accumulator, EMF of a cell, standard electrode potential, Nernst equation and its application to chemical cells, fuel cells; corrosion. Chemical Kinetics Rate of a reaction (average and instantaneous), factors affecting rates of reaction; concentration, temperature, calatyst; order and molecularity of a reaction; rate law and specific rate constant, integrated rate equations and half life (only for zero and first order reactions); concept of collision theory (elementary idea, no mathematical treatment) 36 Surface Chemistry Adsorption - physisorption and chemisorption; factors affecting adsorption of gases on solids; catalysis: homogenous and heterogeneous, activity and selectivity: enzyme catalysis; colloidal state: distinction between true solutions, colloids and suspensions; lyophilic, lyophobic, multimolecular and macromolecular colloids: properties of colloids; Tyndall effect, Brownian movement, electrophoresis, coagulation; emulsion - types of emulsions. General Principles and Processes of Isolation of Elements Principles and methods of extraction - concentration, oxidation, reduction electrolytic method and refining; occurrence and principles of extraction of aluminium, copper, zinc and Iron. p-Block Elements Group - 5 elements : General introduction, electronic configuration, occurrence, oxidation states, trends in physical and chemical properties; nitrogen - preparation, properties and uses; compounds of nitrogen: preparation, properties and uses; compounds of nitrogen: preparation and properties of ammonia and nitric acid, oxides of nitrogen (structure only); Phosphorous-allotropic forms; compounds of phosphorous: preparation and properties of phosphine, halides (PCl3, PCl5) and oxoacids (elementary idea only). Group - 6 elements : General introduction, electronic configuration, oxidation states, occurrence, trends in physical and chemical properties; dioxygen: preparation, properties and uses; simple oxdides; Ozone, Sulphur-allotropic forms; compounds of sulphur: preparation, properties and uses of sulphur dioxide; sulphuric acid: industrial process of manufacture, properties and uses, oxoacids of sulphur (structures only). Group - 7 elements : General introduction, electronic configuration, oxidation states, occurrence, trends in physical and chemical properties; compounds of halogens; preparation, properties and uses of chlorine and hydrochloric acid, interhalogen compounds, oxoacids of halogens (structures only). Group - 8 elements : General introduction, electronic configuration. Occurrence, trends in physical and chemical properties, uses. (Part-II) d- and f- Block Elements General introduction, electronic configuration, occurrence and characteristics of transition metals, general trends in properties of the first row transition metals - metallic character, ionization enthalpy, oxidation states, ionic radii, colour, catalytic property, magnetic properties, interstitial compounds, alloy formation. Preparation and properties of K2Cr2O7 and KMnO4. Lanthanoids - electronic configuration, oxidation states, chemical reactivity and lanthanoid contraction. Actinoids - Electronic configuration, oxidation states. Coordination Compounds Coordination Compounds - Introduction, ligands, coordination number, colour, magnetic properties and shapes, IUPAC nomenclature of mononuclear coordination compounds, bonding; isomerism, importance of coordination compounds (in qualitative analysis, extraction of metals and biological systems). Haloalkanes and Haloarenes Haloalkanes: Nomenclature, nature of C-X bond, physical and chemical properties, mechanism of substitution reactions. Haloarenes: Nature of C-X bond, substitution reactions (directive influence of halogen for monosubstituted compounds only) Uses and environmental effects of - dichloromethane, trichloromethane, tetrachloromethane, iodoform, freons, DDT. Alcohols, Phenols and Ethers Alcohols : Nomenclature, methods of preparation, physical and chemical properties (of primary alcohols only); identification of primary, secondary and tertiary alcohols; mechanism of dehydration, uses of methanol and ethanol. Phenols : Nomenclature, methods of preparation, physical and chemical properties, acidic nature of phenol, electrophillic substitution reactions, uses of phenols. Ethers : Nomenclature, methods of preparation, physical and chemical properties, uses. Aldehydes, Ketones and Carboxylic acids Aldehydes and Ketones : Nomenclature, nature of carbonyl group, methods of preparation, physical and chemical properties mechanism of nucleophilic addition, reactivity of alpha hydrogen in aldehydes; uses. Carboxylic Acids : Nomenclature, acidic nature, methods of preparation, physical and chemical properties; uses. Organic compounds containing Nitrogen Amines : Nomenclature, classification, structure, methods of preparation, physical and chemical properties, uses, identification of primary, secondary and tertiary amines. Cyanides and Isocyanides - will be mentioned at relevant places in context. 37 Diazonium salts : Preparation, chemical reactions and importance in synthetic organic chemistry. Biomolecules Carbohydrates - Classification (aldoses and ketoses), monosaccahrides (glucose and fructose), oligosaccharides (sucrose, lactose, maltose), polysaccharides (starch, cellulose, glycogen); importance. Proteins - Elementary idea of a - amino acids, peptide bond, polypeptides proteins, primary structure, secondary structure, tertiary structure and quaternary structure (qualitative idea only), denaturation of proteins; enzymes. Vitamins - Classification and functions. Nucleic Acids: DNA & RNA. Polymers Classification - natural and synthetic, methods of polymerization (addition and condensation), copolymerization. Some important polymers: natural and synthetic like polythene, nylon, polyesters, bakelite, rubber. Chemistry in Everyday life: 1. Chemicals in medicines - analgesics, tranquilizers, antiseptics, disinfectants, antimicrobials, antifertility drugs, antibiotics, antacids, antihistamines. 2. Chemicals in food - preservatives, artificial sweetening agents. 3. Cleansing agents - soaps and detergents, cleansing action.

MATHEMATICS

SETS AND FUNCTIONS 1. Sets: Sets and their representations, Empty set, Finite & Infinite sets. Equal sets, Subsets, Subsets of the set of real number especially intervals (with notations). Power set, Universal set, Venn diagrams, Union and Intersection of sets, Difference of sets, Complement of a set. 2. Relation & Functions: Ordered pairs, Cartesian product of sets, Number of elements in the Cartesian product of two finite sets. Cartesian product of the reals with itself (upto R X R X R). Definition of relation, pictorial diagrams, domain, condomain and range of a relation. Function as a special kind of relation from one set to another. Pictorial representation of a function, domain, co-domain & range of a function. Real valued function of the real variable, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum and greatest integer functions with their graphs, Sum, difference, product and quotients of functions. 3. Trigonometric Functions: positive and negative angles. Measuring angles in radians & in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identify sin2x + cos2 x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x + y) and cos (x + y) in terms of sin x, sin y, cos x, & cos y. Identities related to sin2x, cos2x, tan2x, sin3x, cos3x and tan3x. General solution of trigonometric equations of the type sin? = sina, cos? = cosa and tan? = tana. Proof and simple application of sine and cosine formula. ALGEBRA (I) 1. Principle of Mathematical Induction : Processes of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications. 2. Complex Numbers and Quadratic Equations: Need for complex numbers, especially v-1, to be motivated by inability to solve every quadratic equations. Brief description of algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system. 3. Linear Inequalities: Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear, inequalities in two variables. Solution of system of linear inequalities in two variables - graphically. 4. Permutation & Combination : Fundamental principle of counting. Factorial n. (n!) Permutation and combinations, derivation of formulae and their connections, simple applications. 5. Binomial Theorem: History, statement and proof of the binomial theorem for positive integral indices. Pascal's triangle, General and middle term in binomial expansion, simple applications. 6. Sequence & Series: Sequence and Series. Arithmetic progression (A.P.) arithmetic mean (A.M.) Geometric progression (G.P.), general term of a G.P. Sum of n terms of a G.P., geometric mean (G.M.), relation between A.M. and G. M. Sum to n terms of the special series , ?n and ?n2 and ?n3. CO-ORDINATE GEOMETRY (I) 1. Straight Lines : Brief recall of 2D from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line : parallel to axes, point-slope form, slope-intercept form, two-point form, intercepts form and normal form. General equation of a line, Distance of a point from a line. 2. Conic Section : Sections of a cone : circle, ellipse, parabola, hyperbola, a point, a straight line and pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle. 39 3. Introduction to Three Dimensional Geometry : Coordinate axes and coordinate planes in three dimensions, Coordinates of a point. Distance between two points and section formula. CALCULUS (I) 1. Limits and Derivatives : Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit. Definition of derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions. MATHEMATICAL REASONING 1. Mathematical Reasoning Mathematically acceptable statements. Connecting words / phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and / or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words - difference between contradiction, converse and contrapositive. STATISTICS & PROBABILITY (I) 1. Statistics : Measure of dispersion; mean deviation, variance and standard deviation of ungrouped / grouped data. Analysis of frequency distributions with equal means but different variances. 2. Probability : Random experiments: outcomes, sample spaces (set representation). Events: occurrence of events, 'not', 'and' 'or' events, exhaustive events, mutually exclusive events Axiomatic (set theoretic) probability, connections with the theories of earlier classes. Probability of an event, probability of 'not' 'and' & 'or' events. * RELATIONS AND FUNCTIONS. 1. Relations and Functions : Types of relations : reflexive, symmetric, transitive and equivalence relations, One to one and onto functions, composite functions inverse of a function, Binary operations. 2. Inverse Trigonometric Functions : Definition, range, domain principal value branches. Graphs of inverse trigonometric functions Elementary properties of inverse trigonometric functions. ALGEBRA (II) 1. Matrices : Concept, notation, order, equality types of matrices zero matrix transpose of a matrix, symmetric and skew symmetric matrices, Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Non-commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations, Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries). 2. Determinants : Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix. CALCULUS (II) 1. Continuity and Differentiability : Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concept of exponential and logarithmic functions and their derivatives. Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives, Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretations. 2. Applications of Derivatives : Applications of derivatives : rate of change, increasing/decreasing functions, tangents & normals, approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principle and understanding of the subject as well as real-life situations). 40 3. Integrals : Integration as inverse process of differentiation. Integration of a variety of functions by substitution by partial fractions and by parts, only simple integrals. Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals. 4. Applications of the Integrals : Applications in finding the area under simple curves, especially lines, areas of circles / parabolas / ellipses (in standard form only), area between the two above said curves (the region should be clearly identifiable). 5. Differential Equations : Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation of variables, homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type: dy ____ + py = q, where p and q are functions of x. dx VECTORS AND THREE DIMENSIONAL GEOMETRY (II) 1. Vectors : Vectors and scalars, magnitude and direction of a vector. Direction cosines / ratios of vectors. Types of vectors (equal unit zero parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors. 2. Three - dimensional Geometry : Direction cosines / ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane. LINEAR PROGRAMMING 1. Linear Programming : Introduction, definition of related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constaints). PROBABILITY (II) 1. Probability : Multiplication therorem on probability. Conditional probability, independent events, total probability, Baye's theorem, Random variable and its probability distribution, mean and variance of haphazard variable. Repeated independent (Bernoulli)trials and Binomial distribution.