Uni-GAUGE 2027 Exam: Dates, Application Form, Syllabus, Eligibility, Pattern, Preparation Tips

Physics (class-XII): Unit 01

• Electric charges; 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

Physics (class-XII): Unit 02

• 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 nonlinear), electrical energy and power
• Electrical resistivity and conductivity
• 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

Physics (class-XII): Unit 03

• 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
• 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
• 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 a magnetic dipole (bar magnet) in a 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

Physics (class-XII): Unit 04

• Electromagnetic induction; Faraday’s laws, induced EMF and current; Lenz’s law, 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, wattless current
• AC generator and transformer

Physics (class-XII): Unit 05

• 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

Physics (class-XII): Unit 06

• 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 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: Microscopes and astronomical telescopes (reflecting and refracting) and their magnifying powers
• Wave optics: Wavefront and Huygens principle, reflection and refraction of plane wave at a plane surface using wavefronts. Proof of laws of reflection and refraction using Huygens Principle
• Wave optics: 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, microscopes and astronomical telescopes
• Wave optics: Polarisation, plane polarised light, Brewster's law, uses of plane polarised light and polaroids

Physics (class-XII): Unit 07

• Dual nature of radiation
• Photoelectric effect, 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)

Physics (class-XII): Unit 08

• Alpha-particle scattering experiment; Rutherford’s model of atom; Bohr model, energy levels, hydrogen
• 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, nuclear fusion

Physics (class-XII): Unit 09

• Energy bands in solids (qualitative ideas only) conductors, insulator 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

Physics (class-XII): Unit 10

• 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 (class-XI and XII): Unit 01

• Introduction to structure of atom: Dalton’s atomic theory
• Atomic models: Thomson model, Rutherford model, Bohr model, dual behavior of matter
• Quantum mechanical model: Concept of orbitals, Heisenberg's Uncertainty principle, quantum numbers
• Shapes of atomic orbitals: Shape of s, p, and d orbitals, node and nodal surface, shielding effect
• Rules for filling electrons in orbitals: Aufbau principle, Pauli’s exclusion principle, Hund’s rule, electronic configuration of atoms
• Stability of completely filled and half-filled orbitals

Chemistry (class-XI and XII): Unit 02

• Types of chemical: Ionic bond, covalent bond, polar covalent bond
• Valence bond theory: Hybridization, VSEPR theory, resonance
• Molecular orbital theory: Magnetic characteristics, bond order
• Hydrogen bond: Intermolecular hydrogen bonding, intramolecular hydrogen bonding

Chemistry (class-XI and XII): Unit 03

• Intermolecular forces: Types of intermolecular forces, nature of intermolecular forces
• Laws governing gaseous state: Boyle’s law, Charles law, Gay-lussac law, Avogadro law
• Ideal behaviour: Ideal gas equation, Dalton’s law of partial pressure, kinetic theory of gases pressure
• Deviation from ideal behaviour: Compressibility factor, Boyle’s temperature
• Liquefaction of gases: Critical temperature, critical pressure and critical volume
• Liquid state: Vapour pressure, viscosity, surface tension

Chemistry (class-XI and XII): Unit 04

• Thermodynamic terms: Concepts of-system, surrounding, types of system, state of a system, state function and path function, extensive and intensive properties, reversible and irreversible process
• Thermodynamic quantities: Work, heat
• First law of thermodynamics: Internal energy, enthalpy, heat capacity, measurement of ∆U, measurement of ∆H
• Thermochemistry: Enthalpy change in a chemical reaction, endothermic and exothermic reactions, standard enthalpy of reactions, enthalpy changes during phase transformations, standard enthalpy of formation, thermochemical equations
• Thermochemistry: Hess's law of constant heat summation, enthalpies for different types of reactions
• Spontaneity: Entropy, second law of thermodynamics, Gibb's energy change for spontaneous and non-spontaneous processes, criteria for equilibrium
• Third law of thermodynamics

Chemistry (class-XI and XII): Unit 05

• Introduction to equilibrium: Dynamic nature of equilibrium
• Equilibrium in physical processes: Solid-liquid equilibrium, liquid-vapour equilibrium, solid vapour equilibrium, equilibrium involving dissolution of solid and gases in liquids
• Equilibrium in chemical processes: Dynamic nature of chemical equilibrium, law of chemical equilibrium, equilibrium constant
• Types of chemical equilibria: Homogeneous equilibria, heterogeneous equilibria
• Applications of equilibrium constant: Predicting the extent of a reaction, predicting the direction of the reaction, calculating equilibrium concentrations
• Factors affecting equilibria: Le Chatelier’s principle
• Ionic equilibrium in solution: Strong and weak electrolytes, acids, bases, and salts
• Ionization of acids and bases: Ionic product of water, pH scale, ionization constant of weak acids and bases, factors affecting acid strength, common ion effect
• Buffer solutions: Buffer action and relevant examples
• Solubility equilibria of sparingly soluble salts: Solubility product, common ion effect of solubility of ionic salts

Chemistry (class-XI and XII): Unit 06

• Introduction to solid state chemistry: Characteristics of solid state
• Classification of solids on the basis of order in the arrangement: Crystalline and amorphous solids
• Crystal lattices and unit cells: Primitive and centred unit cells, number of atoms in per unit cell in a cubic unit cell
• Close packing in solids: Packing in solids, voids, packing efficiency, calculation of density of unit cell
• Imperfections in solid: Types of point defects, stoichiometric and non-stoichiometric defects, metal excess defect, metal deficiency defect, impurity defects
• Electrical properties: Conductors, semiconductors and insulators, band theory of solids, n and p type semiconductors
• Magnetic properties: Paramagnetic, diamagnetic, ferromagnetic, antiferromagnetic, ferrimagnetic

Chemistry (class-XI and XII): Unit 07

• Introduction to solutions: Solute, solvent, solution
• Types of solutions: Gaseous solutions, liquid solutions, solid solutions
• Expressing the concentration of solutions of solids in liquids: Various quantities used to express concentration of a solution, mole fraction, molarity, molality
• Solubility: Solubility of solid in liquid, solubility of gas in liquid, Henry’s law
• Vapour pressure of liquid solutions: Solution of two volatile liquids, solution containing non-volatile solute, Raoult’s law
• Classification of liquid-liquid solutions on the basis of Raoult’s law: Ideal solutions, non ideal solutions, positive deviation, negative deviation
• 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: Van’t Hoff factor-numericals based on the above

Chemistry (class-XI and XII): Unit 08

• Oxidation and reduction reactions
• Redox reactions in terms of electron transfer reactions: Mechanism of redox reactions by electron transfer process, evolution of the electrochemical series
• Oxidation number: Calculation of oxidation number
• Types of redox reactions
• Balancing of redox reactions: Oxidation number method, half reaction, method
• Types of electrochemical cells: Electrolytic cells, galvanic cells
• Electrolysis: Electrode sign conventions at anode and cathode, laws of electrolysis
• Conductance in electrolytic solutions: Metallic and electrolytic conductance, types of electrolytes, conductance resistance, molar conductivity, variation of conductivity with concentration, Kohlrausch’s law
• Galvanic cells: 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
• Corrosion: Concept and mechanism of corrosion in relation to EMF

Chemistry (class-XI and XII): Unit 09

• s-block elements-group 1 elements and group 2 elements: Electronic configuration, physical properties, chemical properties, position of hydrogen in the periodic table, diagonal relationship, biological importance, water and hydrogen peroxide
• s-block elements-group 1 elements and group 2 elements: Some alkali metal compounds, some alkaline earth metal compounds
• p-block elements-group 13, 14, 15, 16, 17, and 18 elements: Electronic configuration, occurrence inert pair effect reactivity, some compounds of group 13 to 18 elements

Chemistry (class-XI and XII): Unit 10

• d-Block elements: General properties of 3d elements. Electronic configuration, variable valency concept color, magnetic properties, catalytic properties, compounds
• f-block elements: Electronic configuration, oxidation states, lanthanide contraction
• Coordination compounds: General composition, coordination number, types of ligands, Werner theory
• IUPAC nomenclature of coordination compounds: IUPAC rules
• Valence bond theory as applied to coordination compounds: IUPAC rules, valence bond theory, crystal field theory
• Importance of coordination compounds: Analytical applications, industrial applications, biological applications

Chemistry (class-XI and XII): Unit 11

• Adsorption on a surface: Physisorption, chemisorption, factors affecting the adsorption of gases on solids
• Catalysis: Homogeneous and heterogeneous catalysis, shape selective catalysis, enzyme catalysis
• Colloids: Distinction between true solution, colloid and suspension, classification of colloids, properties of colloids: Mechanical, optical, electrical, Hardy-Schulze rule, application of colloids

Chemistry (class-XI and XII): Unit 12

• Rate of chemical reaction: Average rate of reaction, instantaneous rate of reaction
• Factors affecting rate of a reaction: Concentration of reactants, temperature, catalyst, nature of reactants, pressure (gases), presence of light, surface area of the reactants rate law and specific rate constant order and molecularity
• Integrated rate equations and half life: Zero order reactions, first order reactions, pseudo first order reaction
• Temperature dependence of rate of reaction: Activation energy, Arrhenius equation
• Collision theory

Chemistry (class-XI and XII): Unit 13

• Types of hybridization of carbon: Types of hybridization in carbon compounds, shapes of organic molecules, 2D and 3D structural representation of organic compounds
• Classification of organic compounds: Based on functional groups, based on structure
• IUPAC nomenclature of organic compounds: Priority order of functional groups, prefixes and suffixes for functional groups, derivation of structural formula from a given IUPAC name and vice-versa
• Stereochemistry and isomerism: Structural isomerism, stereochemistry and stereoisomerism, projection formulae, interconversion of projection formulas, conformations and their relative stabilities (ethane and butane)
• Stereochemistry and isomerism: Geometrical isomerism (cis and trans), optical isomerism, absolute and relative nomenclature of optical isomers
• Homolytic and heterolytic fission of a covalent bond: Carbocation, carbanion, free radical
• Basics of organic reaction: Electrophilic and nucleophilic reagents, types of organic reactions
• Electronic displacements in a covalent bond: Inductive effect, electromeric effect, resonance, hyperconjugation
• Aromaticity: Stability of aromatic compounds, Huckel’s rule
• Alkanes (upto 5 carbon atoms): Methods of preparation (reduction, Wurtz reaction, Kolbe's electrolysis), physical properties, chemical reactions (halogenation, isomerisation, oxidation, aromatization, combustion, pyrolysis)
• Alkenes (upto 5 carbon atoms): Methods of preparation (partial reduction, dehydrohalogenation, dehydration, dehalogenation), physical properties, chemical reactions (addition of H2, X2, Markovnikov’s and anti-Markovnikov’s rule)
• Alkenes (upto 5 carbon atoms): Addition of HX, and HO, ozonolysis, oxidation and polymerization
• Alkynes (upto 5 carbon atoms): Methods of preparation (hydrolysis of calcium carbide, dehydrohalogenation), physical properties, chemical reactions (addition of H2, X2, HX, and HO and polymerization)
• Arenes: Nomenclature, resonance and stability of benzene, orientation effect of substituents in benzene, preparation physical and chemical properties of benzene
• Haloalkanes and haloarenes: Structure classification, structure of 1⁰, 2⁰, and 3⁰ haloalkanes and haloarenes, nomenclature, isomerism, preparation and properties

Chemistry (class-XI and XII): Unit 14

• Structure: Structure of alcohols, phenols, and ethers, classification
• Preparation of alcohols and phenols: Preparation of alcohols (hydration of alkenes, hydroboration-oxidation, reduction of carbonyl compounds, from Grignard's reagent), preparation of phenols (from chlorobenzene, benzene, and cumene)
• Properties of alcohols, phenols, and ethers: Physical properties of alcohols, phenols, and ethers, chemical properties of alcohols (with metals, esterification, esterification, with HX, dehydration)
• Properties of alcohols, phenols, and ethers: Chemical properties of phenols (halogenation, nitration and sulphonation, Kolbe's Reimer-Tiemann, deoxygenation and oxidation)
• Preparation of ethers and chemical properties: Preparation from alcohols, Williamson's ether synthesis, ether cleavage by HX, halogenation, nitration and Friedel Crafts reaction
• Structure of aldehydes, ketones, and carboxylic acids
• Preparation of aldehydes and ketones: From alcohols, from alkenes, from alkynes, from aromatic hydrocarbons, Gattermann-Koch, from acid chlorides, from nitriles
• Physical, chemical properties and uses of aldehydes and ketones: Physical properties of aldehydes and ketones, chemical properties of aldehydes and ketones (nucleophilic addition reactions, nucleophilic addition-elimination reactions, reduction, oxidation
• Physical, chemical properties and uses of aldehydes and ketones: Aldol condensation, Cannizzaro reaction, electrophilic substitution in aromatic aldehydes)
• Carboxylic acids: Structure of carboxylic acid, preparation of carboxylic acids (by oxidation, hydrolysis, from Grignard reagents), physical properties of carboxylic acids, chemical properties of carboxylic acids

Chemistry (class-XI and XII): Unit 15

• Structure
• Preparation of amines: By reduction of nitro compounds, nitriles and amides, ammonolysis of alkyl halides
• Physical and chemical properties of amines: Physical properties of amines, chemical properties of amines
• Diazonium salts: Nomenclature structure, methods of preparation, physical properties, chemical properties structure and importance of azo dyes and examples

Chemistry (class-XI and XII): Unit 16

• Biomolecules: Carbohydrates, amino acids and proteins, nucleic acids, vitamins
• Polymers: Classification, methods of polymerization, preparation of some polymers

Chemistry (class-XI and XII): Unit 17

• Chemicals in medicines, food and hygiene (soaps and detergents): Antacids, antihistamines, tranquilizers, analgesics, antimicrobials (antibiotics, antiseptics and disinfectants), antifertility drugs and chemotherapy, food additives
• Chemicals in medicines, food and hygiene (soaps and detergents): Artificial sweetening agents, preservatives and antioxidants, saponification, soaps and cleansing property, detergents and biodegradable detergents

Chemistry (class-XI and XII): Unit 18

• Environmental: Environmental pollution
• Pollution: Conservation of natural resources
• Water pollution: Types of water pollutants, treatment of water pollution, BOD
• Industrial pollution: Industrial and agricultural, chemicals that, cause environmental degradation, industrial waste management, green chemistry

Mathematics (class-XI): Unit 01

• Sets and their representations, empty set, finite and infinite sets
• Equivalent and 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 on union and intersection of sets

Mathematics (class-XI): Unit 02

• Ordered pairs, Cartesian product of sets
• Number of elements in the cartesian product of two finite sets
• Cartesian product of the set of all reals with itself (upto RxRxR)
• Definition of relation, pictorial diagrams, domain, co-domain 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 valued functions, domain and range of the functions: Constant, identity, linear and quadratic polynomial, rational, modulus, signum and greatest integer functions with their graphs
• Sum, difference, product and quotients of functions, even and odd function

Mathematics (class-XI): Unit 03

• Positive and negative angles
• Measuring angles in radians and in degrees and conversion from one measure to another
• Definition of trigonometric functions with the help of unit circle
• Truth of the identity sin²x +cos²x=1, for all x
• Signs of trigonometric functions
• Domain and range of trigonometric functions and their graphs
• Trigonometric functions as periodic functions, their amplitude, argument period and graph
• Expressing sin (x±y) and cos(x±y) in terms of sin x, sin y, cos x and cos y
• Deducing identities like the following: tan (x ± y) = tan x ± tan y/ 1 ± tan x tan y, cot (x+y) = cot x cot y ± 1/ cot y ± cot x, sin x + sin y = 2 sin x + y/ 2 cos x – y/ 2, cos x + cos y = 2 cos x + y/ 2. cos x – y/ 2
• Deducing identities like the following: sin x – sin y = 2 cos x + y/ 2. sin x – y/2, cos x – cos y = 2 sin x – y/ 2. sin x- y/ 2
• Identities related to Sin2x, Cos2x, tan2x, Sin3x, Cos3x, and tan3x
• General solution of trigonometric equations of the type sinӨ =sinӨ , cosӨ =cosӨ and tanӨ =tanӨ
• Proof and simple application of sine and cosine rules only, law of sine, law of cosine and their applications

Mathematics (class-XI): Unit 04

• 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

Mathematics (class-XI): Unit 05

• Need for complex numbers, especially,to be motivated by inability to solve some of the quadratic equations, standard form of a complex number
• Algebraic properties of complex numbers, Argand plane, the modulus and conjugate of a complex number 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
• Cube roots of unity and their properties

Mathematics (class-XI): Unit 06

• Linear Inequalities
• Algebraic solutions of linear inequalities in one variable and their representation on the number line
• Graphical solution of linear inequality in two variables
• Graphical solution of system of linear inequalities in two variables
• Inequalities involving modulus function
• Practical problems on linear inequality, algebraic solution of quadratic inequality

Mathematics (class-XI): Unit 07

• Fundamental principle of counting
• Factorial n.(n!), permutations and combinations
• Properties of combination, derivation of formulae and their connections, simple applications

Mathematics (class-XI): Unit 08

• Positive integral indices
• General and middle term in binomial expansion, simple applications

Mathematics (class-XI): Unit 09

• Sequences 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., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M.
• Sum to n terms of the special series, and arithmetic geometric series ∑n, ∑n², and ∑n³
• Harmonic progression

Mathematics (class-XI): Unit 10

• 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 axes, point slope form, slope intercept form, two-point form, intercept form and normal form
• 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, distance between parallel lines

Mathematics (class-XI): Unit 11

• Identify the circle, parabola, ellipse and hyperbola as cross sections of a double napped cone by a plane

Mathematics (class-XI): Unit 12

• Coordinate axes and coordinate planes in three dimensions
• Coordinates of a point in space
• Distance between two points and section formula, direction cosines of a line, direction ratios of line, angle between two lines

Mathematics (class-XI): Unit 13

• Intuitive idea of limit of a function
• Derivative introduced as rate of change of distance function and its geometric meaning, definition of derivative, relate it to slope of tangent of the curve, derivative of sum difference, product and quotient of function
• Derivative of polynomials and trigonometric function

Mathematics (class-XI): Unit 14

• Random experiments: Outcomes, sample spaces (set events, exhaustive)
• Events: Occurrence of events, ‘not’, ‘and‘, and ‘or’ event, exclusive events, mutually exclusive event, axiomatic (set theoretic) probability, connections with the theories of earlier classes
• Probability of an event, probability events

Mathematics (class-XII): Unit 01

• Types of relations: Reflexive, symmetric, transitive and equivalence relations
• One to one and onto function, composite functions, inverse of a function
• Binary operations
• Concept of exponential and logarithmic function to the base e, logarithmic function as inverse of exponential function and graphs

Mathematics (class-XII): Unit 02

• Definition of inverse trigonometric function in unit circles, range, domain principal value branches
• Graphs of inverse trigonometric functions, elementary properties of inverse trigonometric functions

Mathematics (class-XII): Unit 03

• Concept, notation, order, equality, types of matrices: Zero matrix, transpose of 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 matrix and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of orders)
• 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)

Mathematics (class-XII): Unit 04

• Determinant of a square matrix (upto 3 x 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle, collinearity of points
• 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)
• Cramer’s rule and its applications on word problems

Mathematics (class-XII): Unit 05

• Adjoint of a square matrix of order 2 2 and 3 3
• Properties of adjoint of a matrix
• Inverse of a square matrix
• Consistency, in consistence and number of solutions of a system of linear equations by examples
• Solving the system of linear equations in two and three variables by matrix method and its application in word problem

Mathematics (class-XII): Unit 06

• Differentiability, derivative introduced as rate of change of distance function and its geometric meaning, definition of derivative relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of function
• Derivative of polynomials and trigonometric function
• Derivative of composite functions, chain rule, derivatives of inverse trigonometric functions derivative of implicit functions
• Derivatives of logarithmic and exponential functions
• Logarithmic differentiation derivative of functions expressed in parametric forms seconds order derivatives

Mathematics (class-XII): Unit 07

• Applications of derivatives rate of change increasing/ decreasing tangents and normal approximation, maxima and minima (first derivative test, local maxima/ local minima and second derivative test, absolute maxima/ absolute minima)
• Simple problems (that illustrate basic principles and understanding to the subject as well as real-life situations)

Mathematics (class-XII): Unit 08

• Integration as inverse process of differentiation, integration of a variety of functions by substitution
• Integration by parts, integration by partial fractions

Mathematics (class-XII): Unit 09

• Fundamental theorem of calculus (without proof), definite integral by substitution, basic properties of definite integrals and evaluation of definite integrals, definite integrals as a limit of sum

Mathematics (class-XII): Unit 10

• Applications in finding the area bounded by a curve and a line
• Area bounded between lines
• Areas bounded between two curves
• Areas of circles/ ellipses (in standard form only)
• Area under the curve y = sinx, y = cosx (the region should be clearly identifiable)

Mathematics (class-XII): Unit 11

• Definition order and degree, general and particular solutions of a differential equation, formation of differential equation whose general solution is given, solution is 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/dx + py = q, where p and q are function x and dx/dy + px = q, where p and q are function of y

Mathematics (class-XII): Unit 12

• Vectors and scalars, magnitude and direction of a vector
• Representation of vectors, types of vectors, position vector of a point, components of a vector, addition of vectors (properties of addition, laws of addition), 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, scalar triple product

Mathematics (class-XII): Unit 13

• Direction cosines and direction 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 two lines
• Angle between two planes
• Angle between a line and a plane
• Distance of a point from a plane

Mathematics (class-XII): Unit 14

• Multiplication theorem on probability
• Conditional probability, variable and its probability distribution, mean and variance of random variable
• Repeated independent (Bernoulli) trials and Binomial distribution