**College / University:**Tolani Maritime Institute (TMI), Mumbai

Tolani Maritime Institute is one of the largest maritime educational centres offering Marine Engineering and Nautical Technology degree programs. TMI graduates are awarded B.Tech. Marine Engineering and B.Tech. Nautical Technology degrees by the well established Birla Institute of Technology and Science (BITS) Pilani, a deemed university widely recognised as one of India’s pre-eminent centres for technology education. Candidates can check the complete details of Tolani Maritime Institute of Science Aptitude Test (TMISAT) Syllabus and other information on this page below.

## 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; SI units; fundamental and derived units. Length; mass and time measurements; accuracy and precision of measuring instruments; errors in measurement; significant figures.

Dimensions of physical quantities; dimensional analysis and its applications.

#### Kinematics:

The frame of reference; Motion in a straight line: Position-time graph; speed and velocity.

Elementary concepts of differentiation and integration for describing motion. 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)

Scalar and vector quantities; Position and displacement vectors; general vectors and their notations; 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. Scalar and vector products of vectors

Motion in a plane. Cases of uniform velocity and uniform acceleration – projectile motion. Uniform circular motion.

#### Laws of Motion:

Intuitiǀe conept of force. Inertia, Newton’s first law of motion; momebtum and Newton’s second law of motion; inpulse; Newton’s third laǁ 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 (vehicle on a level circular road, vehicle on banked road)

## Work, Energy and Power:

Work was 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; motion in a vertical circle; elastic and inelastic collisions in one and two dimensions.

Motion of System of Particles and Rigid Body:

Centre of the mass of a two-particle system; momentum conservation and centre of mass motion.

Centre of a rigid body; centre of mass of a uniform rod

Moment of a force; torque-angular momentum; laws of conservation of angular momentum and its applications.

Equilibrium of rigid bodies; rigid body rotation and equations of rotational motion; comparison of linear and rotational motions.

Moment of inertia; radius of gyration. Values of moment of inertia; for simple geometrical objects (no derivation). Statement of parallel and perpendicular axes theorems and their applications.

#### Gravitation:

Kepler's laws of platetarǇ motion. The universal law of gravitation

Acceleration due to gravity and its variation with altitude and depth.

Gravitational potential energy and gravitational potential. Escape velocity. Orbital velocity of a satellite.

Geo stationary satellites.

#### Properties of Bulk Matter:

Elastic behavior; Stress – strain relationship; Hooke’s law; Young modulus; bulk modulus; shear modulus of rigidity; Poisson’s ration; elastic energy Pressure due to a fluid column Pascal’s law and its application (hydraulic lift and hyraulic brakes).

Effect of gravity on fluid pressure.

Viscosity Stoke’s law; terminal velocity streamline and turbulent flow; critical velocitǇ. Bernoulli’s theorem and its applications.

Surface energy and surface tension; angle of contact; excess of pressure across a curved surface;

application of surface tension ideas to drops; bubbles and capillary rise.

Heat, temperature, thermal expansion, thermal expansion of solids; liquids and gases; anomalous expansion of water; specific heat capacity; Cp; Cv – calorimetry; change of state – latent heat capacity.

Heat transfer-conduction; convection and radiation; thermal conductivity. Qualitative ideas of Blackbody radiation; Wein’s displacement law; Stefan’s law; Green house effect.

#### Thermodynamics:

Thermal equilibrium and definition of temperature (Zeroth law of thermodynamics). Heat, work and internal energy. First law of thermodynamics. Isothermal and adiabatic processes.

The second law of thermodynamics: reversible and irreversible processes. Heat engine and refrigerator.

Behaviour of Perfect Gases and Kinetic Theory of Gases:

The equation of state of perfect gas; work is done in a compressing gas.

Kinetic theory of gases – assumptions; concept of pressure. Kinetic interpretation of temperature; rms speed of gas molecules; degrees of freedom; law of equipartition of energy (statement only) and multiplication to specific heat capabilities of gases; Concept of meanfree path; Aǀogadro’s Number.

#### Oscillations and Waves:

Periodic motion – time periodic; frequency; displacement as a function of time. Periodic functions. Simple harmonic motion (S.H.M) and its equation; phase; oscillations of a spring restoring force and force constant; energy in S.H.M. kinetic and potential energies; simple pendulum derivation of expression for its time period.

Free; forced and damped oscillations (qualitative ideas only) resonance.

Wave motion: Transverse and longitudinal waves; speed of wave motion. Displacement relation for a progressive wave. Principle of superposition of waves; reflection of waves; standing waves in strings and organ pipes; fundamental mode and harmonics; beats, Doppler effect.

#### Electrostatics:

Electric Charges; Conserǀation of Đharge; Coulomď’s law-force between two points charges; forces between multiple charges; superposition principle and continous charge distribution. Electric field; electric field due to a point charge; electric field lines; electric dipole; electric field due to 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

polarization; 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 Graff 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; color 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 ďridge, meter ďridge.

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 and Magnetism:

Concept of mgnetic 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 electrical 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 magnetic dipole (bar magnet) along its axis and perpendicular to its axis. Torque on a magnetic dipole (bar magnet) in a uniform magnetic field; bar magnet as an equivalent solenoid; magnetic field liŶes; 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:

ElectromagnetiĐ InduĐtion; Faraday's laws, induced emf and Đurrent; Lenz’s laws; EddǇ Đurrents. Self and mutual induction. 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; watt less current.

AC generator and transformer.

#### Electromagnetic Waves:

Basic Idea of displacement current; 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 appliĐatioŶs; optiĐal fiďres; refration at spheriĐal surfaĐes; lenses; this lens formula; lensŵaker’s formula. Magnification; power of a lens; combination of thin lenses in a contact combination of a lens and a mirror. Refraction and dispersion of light through a prism.

Scattering of light – blue colour of 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) using lenses. Microscopes and astromical telescopes (reflecting and refracting) and their magnifying powers.

#### Wave Optics:

Waǀe front and Huygen’s priciple; reflection and refraction of plane wave at a plane surface using wave fronts. Proof of laws of reflection and refraction using Huygen’s principle. Interference Young’s douďle slit experiment and expression for fridge width; oherent sources and sustained interference of light. Diffraction due to a single slit, width of central maximum. Resolving power of microscopes and astronomiĐal telescopes. Polarization; plane polarized light Brewster’s law; uses of plane polarized light and Polaroids.

#### Dual Nature of Matter and Radiation:

Dual Nature of 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. Davisson – Germer experiment(experimental details should be omitted; only conclusion should be explained)

#### Atoms & Nuclei:

Alpha particle scattering experiment; Rutherford’s model of atom; Bohr’s 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; adioactive decay law. Mass energy relation; mass defect; binding energy per nucleon and its variation with mass number; nuclear fission; nuclear fusion.

#### Electronic Devices:

Energy bands in solids (Qualitative ideas only) conductors; insulators and semiconductors; semiconductor diode – I-V characteristics 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; characteristics 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 only); 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.

Nature of matter, laws of Chemical Combination, Dalton's atomiĐ theory: the 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; atomic number; isotopes and isobars. Thomson’s model and its limitations. Rutherford’s model and its limitations; Bohr’s model and its limitations; Concept of shells and sub shells; dual nature of matte and light de Broglie’s relationship, Heisennerg uncertainty Principle; concept of orbitals; quantum numbers; shapes of s, p and d orbitals; rules for filling electrons in orbitals – Aufďau Principle; Pauli’s 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; inert gas radii Ionization enthalpy; electron gain enthalpy; electro negativity; valency. Nomenclature of elements with atomic number greater than 100.

#### Chemical Bonding and Molecular Structure:

Valence electrons; ionic bond; covalent bond; bond parameters; Lewis structure; polar character of covalent bond; covalent character of ionic bond; valence bond theory; resonance; geometry of covalent molecules; VSEPR theory; concept of hybrization; involving s,p and d orbitals and shapes of some simple molecules; molecular orbital theory of homonuclear diatomic molecules (qualitative idea only) hydrogen bond.

#### States of Matter: Gases and Liquids:

Three states of matter; intermolecular interactions; types of bonding; melting and boiling points; role of gas laws in elucidating the Concept of the molecule; Boyle’s law, Charles law; Gay Lussac’s law; Avogarƌo’s law; ideal behaviour; empirical derivation of gas equation; Avogadro’s number; ideal gas equation. Derivation from ideal behaviour; liquefaction of gases; critical temperature; kinetic energy and molecular speeds (elementary idea) liquid state-vapor pressure; viscosity and surface tension (qualitative idea only, no mathematical derivations)

#### Chemical Thermodynamics:

Concepts of System and 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 ; Hess law of constant heat summation; enthalpy of bond dissociation; combustion; formation; atomization; sublimation; phase transition; ionization; solution and dilution.

Second law of thermodynamics (brief introduction) Introduction of entropy as a state function; Gibb's energy Change for spontaneous and Non-spontaneous processes; criteria for equilibrium

Third law of thermodynamics (brief introduction)

#### 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; ionization of poly basic acids; acid strength; concept of pH; Henderson Equation; hydrolysis of salts (elementary idea) buffer solution; solubility product; common ion effect (with illustrative examples)

#### Redox Reactions:

The concept of oxidation and reduction; redox reactions; oxidation number; balancing redox reactions; in terms of loss and gain of electrons and change in oxidation number; applications of redox reactions.

#### 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 Sodium and Potassium; Calcium oxide and calcium carbonate and their industrial uses; biological importance of Magnesium and Calcium

#### p-Block Elements:

General introduction to p-Block Elements

Group 13 Elements: General introduction; electronic configuration; occurrence; variation of properties; oxidation states; trends in chemical reactivity; anomalous behavior of first elements. Carbon-catenation; allotropic forms; physical and chemical properties; uses of some important compounds; oxides. Important compounds of Silicon and a few uses: Silicon Tetrachloride; Silicones; Silicates and Zeolites; their uses.

Organic Chemistry – Some Basic Principles and Techniques:

General introduction; methods of purification; 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. Homolytic and heterolytic fission of a covalent bond; free radicals; carbocations; carbonions; electrophiles and nucleophiles; types of organic reactions.

#### Hydrocarbons:

Classifications of Hydrocarbons

Aliphatic Hydrocarbons:

Alkanes – Nomenclature; isomerism; conformation (ethane only); physical properties; chemical reactions including free radical mechanism of halogenations, combustion and pyrolysis.

Alkenes – Nomenclature; structure of double bond (ethene); geometrical isomerism; physical properties; methods of preparation; chemical reactions: addition of hydrogen; halogen; water; hydrogen halides ;(Markownikov’s addition and perioxcide effects ozonolysis; oxidation; mechanism of electrophilic addition.

Alkynes – Nomenclature, the structure of triple bond (ethyne); physical properties; methods of preparation; chemical reactions; the acidic character of alkynes; the addition of – hydrogen; halogens; hydrogen halides and water.

Aromatic Hydrocarbons: Introduction; IUPAC nomenclature; benzene; resonance; aromaticity; chemical properties: mechanism of electrophilic substitution; nitration, sulphonation; halogenation; Friedel Craft’s alkylation and acylation; directive influence of functional group in monosubstituted benzene; Carcinogenicity and toxicity.

#### Environmental Chemistry:

Environmental pollution – air, water and soil pollution; chemical reactions in atmosphere; smog; 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; strategies 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; packing efficiency; voids; number of atoms per unit cell cubic unit cell; points defects; electrical and magnetic properties. Band theory of metals; conductors; semiconductors and insulators and n&p type semiconductors.

#### Solutions:

Types of solutions; expression of concentration of solution of solids in liquids; solubility of gases in liquids; solids solutions; colligative properties – relatiǀe lowering of vapour pressure; Raoult’s law; elevation of boiling point; depression of freezing point; osmotic pressure; determination of molecular masses using colligative properties; abnormal molecular mass can't Hoff factor.

#### Electrochemistry:

Redox reactions; conductance in electrolytic solutions; specific and molar conductivity; variations of Conductivity with concentration; Kohlrausch’s law; electrolysis acd law of electroysis ;(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; Relation between Gibbs energy change and emf of a cell; fuel cells; corrosion.

#### Chemical Kinetics;

Rate of reaction (Average and instantaneous) factors affecting rate of reaction; concentration; temperature; catalyst; order and molecularity of a reaction, rate law of 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). Activation energy, Arrhenious equation.

#### Surface Chemistry:

Adsorption – physisorption and chemisorptions; factors affecting adsorption of gases on solids; catalysis; homogenous and heterogenous activity and selectivity; enzyme catalysis colloidal state distinction between true solutions; colloids and suspension; 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 15 Elements: General introduction ; electronic configuration; occurrence; oxidation states; trends in physical and chemical properties; nitrogen preparation properties & uses; compounds of nitrogen; preparation and properties of ammonia and nitric acid; oxides of nitrogen (structure only); Phosphorus – allotropic forms; compounds of phosphorus: preparation and properties of phosphine; halides PCI3; PCI5 and oxoacids (elementary idea only)

Group 16 Elements: General introduction; electronic configuration; oxidation states; occurrence; trends in physical and chemical properties ; dioxygen: Preparation, properties and uses; classification of oxides; ozone, sulphure – allotropic forms; compounds of sulphure – allotropic forms; compounds of sulphure; preparation properties and uses of sulphur dioxide; sulphuric acid; industrial process of manufacture; properties and uses; oxoacids of sulphur (structure only)

Group 17 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; interhalogens compounds; oxoacids of halogen (structure only)

Group 18 Elements: General introduction; electronic configuration; occurrence; trends in physical and chemical properties, uses.

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.

Lanthanoids: Electronic configuration; oxidation states; chemical reactivity and lanthanoid contraction and its consequences.

Actinoids – Electronic configuration; oxidation states and comparison with lanthanoids.

#### Coordination Compounds:

Coordination compounds – Introduction; ligands; coordination number; colour; magnetic properties and shapes ; IUPAC Nonmenclature of mononuclear Coordination Compounds. Bonding Werner’s theory VBT and CFT structure and stereo isomerism; importance of coordination compounds in (qualitative inclusion; extraction of metals and biological system)

#### Haloalkanes and Haloarenes:

Haloalkanes: Nomenclature; nature of C-X bond, physical and chemical properties; mechanism of substitution reactions; optical rotation.

Haloarenes: Natures of C – X bond; substitution reactions (Directive influence of halogen in 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 with special reference to methanol and ethanol.

Phenols: Nomenclature; methods of preparation; physical and chemical properties; acidic nature of phenols; electrophilic substitution reaction; 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; mechanical of nucleophilic addition; reactivity of alpha hydrogen in aldehydes: uses.

#### Carboxylic Acids:

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.

#### Diaszonium salts:

Preparation; chemical reactions and importance in synthetic organic chemistry.

#### Biomolecules:

Carbohydrates – classification (aldoses and ketones) monosaccahrides (glucose and fructose), D-L conficuration oligosaccharides (sucrose, lactose, maltose) polysaccharides (starch, cellulose, glycogen) importance.

Proteins – Elementary idea of amino acids, peptide bond, polypeptides, proteins, structure of proteins – primary, secondary, tertiary structure and quaternary structure (qualitative idea only), denaturation of proteins; enzymes. Hormones – Elementary idea excluding structure. Vitamins – Classification and functions.

Nucleic Acids: DNA and 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. Biodegradable and not- biodegradable polymers.

#### Chemistry in Everyday life:

Chemicals in medicines – analgesics, tranquillizers antiseptics; disinfectants; antimicrobials; antifertility drugs; antibiotics; antacids; antihistamines.

Chemicals in food – preservations; artificial sweetening agents; elementary idea of antioxidants

Cleansing Agents- soaps and detergents; cleansing action.

## Mathematics

#### Sets:

Sets and their representations; Empty Sets; Finite and Infinite Sets; Equal sets; Subsets. Subsets of a set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement Sets. Practical problems based on sets.

#### Relations & Functions:

Ordered pairs, Cartesian product of sets, number of elements in the Cartesian product of two finite sets. Cartesian product of the sets of real (upto RxR). Definition of relation, pictorial diagrams, domain, codomain 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 and range of a function. Real value functions, 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.

#### Trigonometric Functions:

Positive and negative angles. Measuring angles in radians and in degrees and conversion of one into another. Definition of trigonometric functions with the help of unit circle. Signs of trigonometric functions. Domain and range of trigonometric functions and their graphs. Expressing sin and cos in terms of sex, cosx & cosy and their simple application. General solution of trigonometric equations of the type.

#### Algebra:

Principles of Mathematical Induction:

Principle of Mathematical Induction:

The process 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.

#### Complex Numbers and Quadratic Equations:

Need for complex numbers, especially to be motivated by inability to solve some of the quadratic equations. 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. Square root of a complex number.

#### Linear Inequalities:

Linear inequalities, algebraic solutions of linear inequalities in one variable and their representations on the number line. Graphical solution of linear inequalities in two variables. Graphical solution of linear inequalities in two variables. Graphical solution of the system of linear inequalities in two variables.

#### Permutations and Combinations:

The fundamental principle of counting. Factorial n Permutations and combinations, derivation of formulae and their connections, simple applications.

#### 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.

#### Sequence and Series:

Sequence and Series. Arithmetic Progression (A.P) Arithmetic Mean (A.M) Geometric Progression and general Term of a G.P. sum of n terms of a G.P. Arithmetic and Geometric series infinite G.P. and its sum; geometric mean; relation between A.M. and G.M. formula

#### Coordinate Geometry:

Straight lines: Brief recall of two-dimensional geometry from earlier classes. Shifting of origin. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis; point slope form, slope intercept form; two point form; intercept form and normal. General equation of a line. Equation of family of lines passing through the point of intersection of two lines. Distance of a point from a line.

Conic Sections: Sections of a cone; circles, ellipse; parabola; hyperbola; a point; a straight line and a 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.

Introduction to Three-dimensional Geometry: Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. The distance between two points and section formula.

#### Calculus:

Limits and Derivatives:

Derivative and introduced as rate of change both as that of distance function and geometrically. Intuitive idea of limit. Limit of polynomials and rational functions, trigonometric; exponential and logarithmic functions. Definitions of derivatives, relate it to slope of tangent of a curve, derivative of sum; difference; product and quotient of functions. The derivative of polynomial and trigonometric functions.

Mathematical Reasoning:

Mathematically acceptable statements. Connecting words/ phrases – consolidating the understanding of " if and only if (ncessary and sufficient) Condition "Implies""and /or implied by " and "or" "their 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 and Probability:

Statistics: Measures of dispersion; mean deviation; variance and standard deviation of ungrouped / grouped data. Analysis of frequency distribution with equal means but different variances.

Probability: Random experiments; outcomes; sample spaces (set representation) events; occurrence of events; "Not" and "Or" events, mutually exclusive events. Axiomatic ; (set theoretic) probability connections with the theories of earlier classes. Probability of ͞"not" "and" and "or" Events.

#### Relations and Functions:

Relations and Functions; Inverse Trigonometric Functions

Algebra:

Matrices; Determinants

Calculus:

Continuity and Differentiability; Applications of Derivatives; Integrals; Applications of the Integrals; Differentials Equation

Vector and Three-Dimensional Geometry:

Vectors; Three-dimensional Geometry

#### Linear Programming:

Introduction; 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; feasibility and infeasible regions, feasible and infeasible solutions; optimal feasible solutions (up to three non-trivial constraints)

#### Probability:

Conditional probability; multiplication theorem on probability; independent events; total probability; Bayes theorem; random variable and its probability distribution; mean and variance of the random variable. Repeated independent (Bernoulli) trials and Binomial distribution.