JEE Main Exam 2025 - Date, Registration, Pattern, Syllabus, Sample Papers, Cutoff

Physics: Unit 01

• Units of measurements, system of units, S.I. units, fundamental and derived units, least count, significant figures, errors in measurements, dimensions of physics quantities, dimensional analysis, and its applications

Physics: Unit 02

• The frame of reference, motion in a straight line, position time graph, speed and velocity, uniform and non-uniform motion, average speed and instantaneous velocity, uniformly accelerated motion, velocity time, position time graph
• Relations for uniformly accelerated motion, scalars and vectors, vector
• Addition and subtraction, scalar and vector products, unit vector, resolution of a vector
• Relative velocity, motion in a plane, projectile motion, uniform circular motion

Physics: Unit 03

• Force and inertia, Newton’s first law of motion; Momentum, Newton’s second law of motion, impulses, 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
• Dynamics of uniform circular motion: Centripetal force and its applications, vehicle on a level circular road, vehicle on a banked road

Physics: Unit 04

• Work done by a constant force and a variable force, kinetic and potential energies, work energy theorem, power
• The potential energy of spring conservation of mechanical energy, conservative and non conservative forces, motion in a vertical circle: Elastic and inelastic collisions in one and two dimensions

Physics: Unit 05

• Centre of the mass of a two-particle system, centre of the mass of a rigid body, basic concepts of rotational motion, moment of a force, torque, angular momentum, conservation of angular momentum and its applications; the moment of inertia
• The radius of gyration, values of moments of inertia for simple geometrical objects, parallel and perpendicular axes theorems, and their applications
• Equilibrium of rigid bodies, rigid body rotation and equations of rotational motion, comparison of linear and rotational motions

Physics: Unit 06

• The universal law of gravitation
• Acceleration due to gravity and its variation with altitude and depth
• Kepler’s law of planetary motion
• Gravitational potential energy; gravitational potential
• Escape velocity, motion of a satellite, orbital velocity, time period, and energy of satellite

Physics: Unit 07

• Elastic behaviour, stress-strain relationship, Hooke's law
• Young's modulus, bulk modulus, and modulus of rigidity
• Pressure due to a fluid column; Pascal's law and its applications
• Effect of gravity on fluid pressure
• Viscosity
• Stokes' law
• Terminal velocity, streamline, and turbulent flow
• Critical velocity
• Bernoulli's principle and its applications
• Surface energy and surface tension, angle of contact, excess of pressure across a curved surface, application of surface tension drops, bubbles, and capillary rise
• Heat, temperature, thermal expansion; specific heat capacity, calorimetry; change of state, latent heat
• Heat transfer conduction, convection, and radiation

Physics: Unit 08

• Thermal equilibrium, zeroth law of thermodynamics, the concept of temperature
• Heat, work, and internal energy
• The first law of thermodynamics, isothermal and adiabatic processes
• The second law of thermodynamics: Reversible and irreversible processes

Physics: Unit 09

• Equation of state of a perfect gas, work done on compressing a gas, kinetic theory of gases assumptions, the concept of pressure
• Kinetic interpretation of temperature, RMS speed of gas molecules: Degrees of freedom
• Law of equipartition of energy and applications to specific heat capacities of gases; mean free path, Avogadro's number

Physics: Unit 10

• Oscillations and periodic motion - time period, 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
• Wave motion
• Longitudinal and transverse waves, speed of the travelling wave
• 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

Physics: Unit 11

• Electric charges: Conservation of charge, Coulomb's law forces 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 a uniform electric field
• Electric flux
• Gauss's law and its applications to find field due to infinitely long uniformly charged straight wire uniformly charged infinite plane sheet, and uniformly charged thin spherical shell
• Electric potential and its calculation for a point charge, electric dipole and system of charges, potential difference, equipotential surfaces, electrical potential energy of a system of two point charges and of electric dipole in an electrostatic field
• Conductors and insulators
• Dielectrics and electric polarization, capacitors and capacitances, the combination of capacitors in series and parallel, and capacitance of a parallel plate capacitor with and without dielectric medium between the plates
• Energy stored in a capacitor

Physics: Unit 12

• Electric current
• Drift velocity, mobility, and their relation with electric current, Ohm's law
• Electrical resistance
• V-l characteristics of Ohmic and non-Ohmic conductors
• Electrical energy and power
• Electrical resistivity and conductivity
• Series and parallel combinations of resistors; temperature dependence of resistance
• Internal resistance, potential difference, and EMF of a cell, a combination of cells in series and parallel
• Kirchhoff’s laws and their applications
• Wheatstone bridge
• Metre bridge

Physics: Unit 13

• Biot - Savart law and its application to the current carrying circular loop
• Ampere's law and its applications to infinitely long current carrying straight wire and solenoid
• Force on a moving charge in uniform magnetic and electric fields
• Force on a current-carrying conductor in a uniform magnetic field
• The force between two parallel currents carrying conductors-definition of ampere
• Torque experienced by a current loop in a uniform magnetic field: Moving coil galvanometer, its sensitivity, and conversion to ammeter and voltmeter
• Current loop as a magnetic dipole and its magnetic dipole moment
• Bar magnet as an equivalent solenoid, magnetic field lines; magnetic field due to a magnetic dipole (bar magnet) along its axis and perpendicular to its axis
• Torque on a magnetic dipole in a uniform magnetic field
• Para-, dia- and ferromagnetic substances with examples, the effect of temperature on magnetic properties

Physics: Unit 14

• Electromagnetic induction, Faraday's law
• Induced EMF and current: Lenz’s law, Eddy currents
• Self and mutual inductance
• Alternating currents, peak and RMS value of alternating current/ voltage: Reactance and impedance: LCR series circuit, resonance: Power in AC circuits, wattless current
• AC generator and transformer

Physics: Unit 15

• Displacement current
• Electromagnetic waves and their characteristics, transverse nature of electromagnetic waves, electromagnetic spectrum (radio waves, microwaves, infrared, visible, ultraviolet X-rays, gamma rays), applications of E.M. waves

Physics: Unit 16

• Reflection of light, spherical mirrors, mirror formula
• Refraction of light at plane and spherical surfaces, thin lens formula, and lens maker formula
• Total internal reflection and its applications
• Magnification
• Power of a lens
• Combination of thin lenses in contact
• Refraction of light through a prism
• Microscope and astronomical telescope (reflecting and refracting ) and their magnifying powers
• Wave optics: Wavefront and Huygens' principle, 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
• Wave optics: Diffraction due to a single slit, width of central maximum, polarization, plane-polarized light, Brewster's law, uses of plane-polarized light and polaroid

Physics: Unit 17

• Dual nature of radiation
• Photoelectric effect
• Hertz and Lenard's observations; Einstein's photoelectric equation: Particle nature of light
• Matter waves - wave nature of particle, De-Broglie relation

Physics: Unit 18

• Alpha particle scattering experiment, Rutherford's model of atom; Bohr model, energy levels, hydrogen spectrum
• Composition and size of nucleus, atomic masses, mass-energy relation, mass defect; binding energy per nucleon and its variation with mass number, nuclear fission, and fusion

Physics: Unit 19

• Semiconductors; semiconductor diode: I-V characteristics in forward and reverse bias; diode as a rectifier, I-V characteristics of LED
• The photodiode, solar cell, and Zener diode; Zener diode as a voltage regulator
• Logic gates (OR. AND. NOT. NAND and NOR)

Physics: Unit 20

• Familiarity with the basic approach and observations of the experiments and activities: (1) Vernier calipers its use to measure the internal and external diameter and depth of a vessel
• Familiarity with the basic approach and observations of the experiments and activities: (2) Screw gauge - its use to determine the thickness/ diameter of thin sheet/ wire
• Familiarity with the basic approach and observations of the experiments and activities: (3) Simple pendulum-dissipation of energy by plotting a graph between the square of amplitude and time
• Familiarity with the basic approach and observations of the experiments and activities: (4) Metre scale-the mass of a given object by the principle of moments. (5) Young's modulus of elasticity of the material of a metallic wire
• Familiarity with the basic approach and observations of the experiments and activities: (6) Surface tension of water by capillary rise and effect of detergents
• Familiarity with the basic approach and observations of the experiments and activities: (7) Coefficient of viscosity of a given viscous liquid by measuring the terminal velocity of a given spherical body
• Familiarity with the basic approach and observations of the experiments and activities: (8) Speed of sound in air at room temperature using a resonance tube. (9) Specific heat capacity of a given - (i) solid and (ii) liquid by method of mixtures
• Familiarity with the basic approach and observations of the experiments and activities: (10) The resistivity of the material of a given wire using a metre bridge. (11) The resistance of a given wire using Ohm's law
• Familiarity with the basic approach and observations of the experiments and activities: (12) Resistance and figure of merit of a galvanometer by half deflection method
• Familiarity with the basic approach and observations of the experiments and activities: (13) The focal length of; (i) convex mirror (ii) concave mirror, and (iii) convex lens, using the parallax method
• Familiarity with the basic approach and observations of the experiments and activities: (14) The plot of the angle of deviation vs angle of incidence for a triangular prism. (15) The refractive index of a glass slab using a travelling microscope
• Familiarity with the basic approach and observations of the experiments and activities: (16) Characteristic curves of a P-N junction diode in forward and reverse bias. (17) Characteristic curves of a zener diode and finding reverse breakdown voltage
• Familiarity with the basic approach and observations of the experiments and activities: (18) Identification of diode. LED, resistor. A capacitor from a mixed collection of such items

Chemistry: Unit 01

• Matter and its nature, Dalton's atomic theory: Concept of atom, molecule, element, and compound, laws of chemical combination; atomic and molecular masses, mole concept, molar mass, percentage composition
• Empirical and molecular formulae: Chemical equations and stoichiometry

Chemistry: Unit 02

• Nature of electromagnetic radiation, photoelectric effect; spectrum of the hydrogen atom
• Bohr model of a hydrogen atom - its postulates, derivation of the relations for the energy of the electron and radius of the different orbits, limitations of Bohr's model; dual nature of matter, De Broglie's relationship.
• Heisenberg uncertainty principle
• Elementary ideas of quantum mechanics, quantum mechanics, the quantum mechanical model of the atom, and its important features
• Concept of atomic orbitals as one-electron wave functions: Variation of Ψ and Ψ² with r for 1s and 2s orbitals; various quantum numbers (principal, angular momentum, and magnetic quantum numbers) and their significance
• Shapes of s, p, and d - orbitals, electron spin, and spin quantum number, rules for filling electrons in orbitals–Aufbau principle
• Pauli's exclusion principle and Hund's rule, electronic configuration of elements, and extra stability of half-filled and completely filled orbitals

Chemistry: Unit 03

• Kossel-Lewis approach to chemical bond formation, the concept of ionic and covalent bonds
• Ionic bonding: Formation of ionic bonds, factors affecting the formation of ionic bonds; calculation of lattice enthalpy
• Covalent Bonding: Concept of electronegativity, Fajan’s rule, dipole moment, valence shell electron pair repulsion (VSEPR ) theory and shapes of simple molecules
• Quantum mechanical approach to covalent bonding: Valence bond theory-its important features, the concept of hybridization involving s, p, and d orbitals; resonance
• Molecular orbital theory: Its important features. LCAO's, types of molecular orbitals (bonding, antibonding), sigma and pi-bonds, molecular orbital electronic configurations of homonuclear diatomic molecules
• Molecular orbital theory: Its important features. The concept of bond order, bond length, and bond energy
• Elementary idea of metallic bonding, hydrogen bonding and its applications

Chemistry: Unit 04

• Fundamentals of thermodynamics: System and surroundings, extensive and intensive properties, state functions, entropy, types of processes
• The first law of thermodynamics: Concept of work, heat internal energy and enthalpy, heat capacity; molar heat capacity; Hess’s law of constant heat summation; enthalpies of bond dissociation, combustion, formation, atomization, sublimation
• The first law of thermodynamics: Phase transition, hydration, ionization, and solution
• The second law of thermodynamics: Spontaneity of processes; ∆S of the universe and ∆G of the system as criteria for spontaneity. ∆G◦ (standard Gibbs energy change) and equilibrium constant

Chemistry: Unit 05

• Different methods for expressing the concentration of solution-molality, molarity, mole fraction, percentage (by volume and mass both), the vapour pressure of solutions and Raoult's Law, ideal and non-ideal solutions
• Vapour pressure-composition, plots for ideal and non ideal solutions; colligative properties of dilute solutions - a relative lowering of vapour pressure, depression of freezing point, the elevation of boiling point and osmotic pressure
• Determination of molecular mass using colligative properties; abnormal value of molar mass, Van’t Hoff factor and its significance

Chemistry: Unit 06

• Meaning of equilibrium is the concept of dynamic equilibrium
• Equilibria involving physical processes: Solid-liquid, liquid-gas - gas and solid-gas equilibria, Henry's law, general characteristics of equilibrium involving physical processes
• Equilibrium involving chemical processes: Law of chemical equilibrium, equilibrium constants (Kp and Kc) and their significance, the significance of ∆G and ∆G◦ in chemical equilibrium
• Equilibrium involving chemical processes: Factors affecting equilibrium concentration, pressure, temperature, the effect of catalyst, Le Chatelier’s principle
• Ionic equilibrium: Weak and strong electrolytes, ionization of electrolytes, various concepts of acids and bases (Arrhenius, Bronsted-Lowry and Lewis) and their ionization, acid-base equilibria (including multi stage ionization) and ionization constants
• Ionic equilibrium: Ionization of water pH scale, common ion effect, hydrolysis of salts and pH of their solutions, the solubility of sparingly soluble salts and solubility products, and buffer solutions

Chemistry: Unit 07

• Electronic concepts of oxidation and reduction, redox reactions, oxidation number, rules for assigning oxidation number, and balancing of redox reactions
• Electrolytic and metallic conduction, conductance in electrolytic solutions, molar conductivities and their variation with concentration: Kohlrausch’s law and its applications
• Electrochemical cells-electrolytic and Galvanic cells, different types of electrodes, electrode potentials including standard electrode potential, half-cell and cell reactions
• EMF of a galvanic cell and its measurement: Nernst equation and its applications; relationship between cell potential and Gibbs' energy change, dry cell and lead accumulator, fuel cells

Chemistry: Unit 08

• Rate of a chemical reaction, factors affecting the rate of reactions: Concentration, temperature, pressure, and catalyst, elementary and complex reactions, order and molecularity of reactions, rate law, rate constant and its units
• Differential and integral forms of zero and first-order reactions, their characteristics and half-lives, the effect of temperature on the rate of reactions, Arrhenius theory, activation energy and its calculation
• Collision theory of bimolecular gaseous reactions (no derivation)

Chemistry: Unit 09

• Modem periodic law and present form of the periodic table, s, p,d and f block elements, periodic trends in properties of elements atomic and ionic radius, ionization enthalpy, electron gain enthalpy, valence, oxidation states, and chemical reactivity

Chemistry: Unit 10

• Group-13 to group 18 elements: General introduction-electronic configuration and general trends in physical and chemical properties of elements across the periods and down the groups; unique behaviour of the first element in each group

Chemistry: Unit 11

• Transition elements: General introduction, electronic configuration, occurrence and characteristics, general trends in properties of the first-row transition elements-physical properties, ionization enthalpy, oxidation states, atomic radii
• Transition elements: General trends in properties of the first-row transition elements-colour, catalytic behaviour, magnetic properties, complex formation, interstitial compounds, alloy formation; preparation, properties, and uses of K2Cr2O7, and KMnO4
• Inner transition elements: (a) Lanthanoids-electronic configuration, oxidation states, and lanthanoid contraction. (b) Actinoids-electronic configuration and oxidation states

Chemistry: Unit 12

• Introduction to coordination compounds
• Werner's theory; ligands, coordination number, denticity
• Chelation; IUPAC nomenclature of mononuclear coordination compounds; isomerism; bonding-valence bond approach and basic ideas of crystal field theory; colour and magnetic properties
• Importance of coordination compounds (in qualitative analysis, extraction of metals, and in biological systems)

Chemistry: Unit 13

• Purification: Crystallization, sublimation, distillation, differential extraction, and chromatography principles and their applications
• Qualitative analysis: Detection of nitrogen, sulphur, phosphorus, and halogens
• Quantitative analysis (basic principles only): Estimation of carbon, hydrogen, nitrogen, halogens, sulphur, and phosphorus
• Calculations of empirical formula and molecular formulae: Numerical problems in organic quantitative analysis

Chemistry: Unit 14

• Tetravalency of carbon: Shapes of simple molecules, hybridization (s and p)-classification of organic compounds based on functional groups, and those containing halogens, oxygen, nitrogen, and sulphur
• Tetravalency of carbon: Homologous series-isomerism-structural and stereoisomerism
• Nomenclature (trivial and IUPAC): Covalent bond fission-homolytic and heterolytic, free radicals, carbocations, and carbanions; stability of carbocations and free radicals, electrophiles, and nucleophiles
• Electronic displacement in a covalent bond: Inductive effect, electromeric effect, resonance, and hyperconjugation
• Common types of organic reactions: Substitution, addition, elimination, and rearrangement

Chemistry: Unit 15

• Classification, isomerism, IUPAC nomenclature, general methods of preparation, properties, and reactions
• Alkanes: Conformations-Sawhorse and newman projections (of ethane), mechanism of halogenation of alkanes
• Alkenes: Geometrical isomerism-mechanism of electrophilic addition: addition of hydrogen, halogens, water, hydrogen halides (Markownikoff's and peroxide effect), ozonolysis and polymerization
• Alkynes: Acidic character-addition of hydrogen, halogens, water, and hydrogen halides, polymerization
• Aromatic hydrocarbons: Nomenclature, benzene structure and aromaticity: Mechanism of electrophilic substitution, halogenation, nitration. Friedel-Crafts alkylation and acylation, directive influence of the functional group in monosubstituted benzene

Chemistry: Unit 16

• General methods of preparation, properties, and reactions; nature of C-X bond; mechanisms of substitution reactions
• Uses; environmental effects of chloroform, iodoform freons, and DDT

Chemistry: Unit 17

• General methods of preparation, properties, reactions, and uses
• Alcohols, phenols, and ethers: (a) Alcohols-identification of primary, secondary, and tertiary alcohols-mechanism of dehydration
• Alcohols, phenols, and ethers: (b) Phenols-acidic nature, electrophilic substitution reactions-halogenation, nitration, and sulphonation. Reimer-Tiemann reaction
• Alcohols, phenols, and ethers: (c) Ethers-structure
• Aldehyde and ketones: Nature of carbonyl group, nucleophilic addition to >C=O group, relative reactivities of aldehydes and ketones; important reactions such as nucleophilic addition reactions (addition of HCN, NH3, and its derivatives), Grignard reagent
• Aldehyde and ketones: Oxidation-reduction (Wolf Kishner and Clemmensen), the acidity of α-hydrogen, aldol condensation, Cannizzaro reaction, haloform reaction, chemical tests to distinguish between aldehydes and ketones
• Carboxylic acids: Acidic strength and factors affecting it

Chemistry: Unit 18

• General methods of preparation, properties, reactions, and uses
• Amines: Nomenclature, classification structure, basic character, and identification of primary, secondary, and tertiary amines and their basic character
• Diazonium salts: importance in synthetic organic chemistry

Chemistry: Unit 19

• General introduction and importance of biomolecules
• Carbohydrates: Classification, aldoses and ketoses, monosaccharides (glucose and fructose) and constituent monosaccharides of oligosaccharides (sucrose, lactose, and maltose)
• Proteins: Elementary idea of α-amino acids, peptide bond, polypeptides, proteins: primary, secondary, tertiary, and quaternary structure (qualitative idea only), denaturation of proteins, enzymes
• Vitamins: Classification and functions
• Nucleic acids: Chemical constitution of DNA and RNA
• Biological functions of nucleic acids
• Hormones (general introduction)

Chemistry: Unit 20

• Detection of extra elements (nitrogen, sulphur, halogens) in organic compounds; detection of the following functional groups; hydroxyl (alcoholic and phenolic), carbonyl (aldehyde and ketones) carboxyl, and amino groups in organic compounds
• The chemistry involved in the preparation of the following: (a) Inorganic compounds; Mohr’s salt, potash alum. (b) Organic compounds: Acetanilide, p-nitro acetanilide, aniline yellow, iodoform
• The chemistry involved in the titrimetric exercises: Acids, bases, and the use of indicators, oxalic-acid vs KMnO4, Mohr’s salt vs KMnO4
• Chemical principles involved in the qualitative salt analysis: (a) Cations-Pb²⁺, Cu²⁺, Al³⁺, Fe³+, Zn²⁺, Ni²⁺, Ca²⁺, Ba²⁺, Mg²⁺, NH₄⁺. (b) Anions-CO²₃¯ , S²¯, SO²₄¯, NO³¯, NO²¯, Cl¯, Br¯, I¯ (insoluble salts excluded)
• Chemical principles involved in the following experiments: (i) Enthalpy of solution of CuSO4, (ii) enthalpy of neutralization of strong acid and strong base, (iii) preparation of lyophilic and lyophobic sols
• Chemical principles involved in the following experiments: (iv) Kinetic study of the reaction of iodide ions with hydrogen peroxide at room temperature

Paper 2A (B.Arch.)-Aptitude test: Unit 01

• Objects, texture related to architecture and build-environment, visualizing three dimensional objects from two-dimensional drawings
• Visualizing
• Different sides of three-dimensional objects
• Analytical reasoning mental ability (visual, numerical, and verbal)

Paper 2A (B.Arch.)-Aptitude test: Unit 02

• Understanding and appreciation of scale and proportions of objects, building forms and elements, colour texture harmony and contrast design and drawing of geometrical or abstract shapes and patterns in pencil
• Transformation of forms both 2D and 3D union, subtraction rotation, development of surfaces and volumes, generation of plans, elevations, and 3D views of objects, creating two-dimensional and three-dimensional compositions using given shapes and forms

Paper 2A (B.Arch.)-Drawing test: Unit 01

• Public space, market, festivals, street scenes, monuments, recreational spaces, etc

Paper 2A (B.Arch.)-Drawing test: Unit 02

• Riverfronts, jungle, gardens, trees, plants, etc

Paper 2B (B.Planning.)-Mathematics: Unit 01

• Sets and their representation: Union, intersection, and complement of sets and their algebraic properties, power set, relation, type of relations, equivalence relations, functions; one-one, into and onto functions, the composition of functions

Paper 2B (B.Planning.)-Mathematics: Unit 02

• Complex numbers as ordered pairs of reals, representation of complex numbers in the form a + ib and their representation in a plane, Argand diagram, algebra of complex number, modulus, and argument (or amplitude) of a complex number
• Quadratic equations in real and complex number system and their solutions, relations between roots and coefficient, nature of roots, the formation of quadratic equations with given roots

Paper 2B (B.Planning.)-Mathematics: Unit 03

• Matrices, algebra of matrices, type of matrices, determinants, and matrices of order two and three, evaluation of determinants, area of triangles using determinants
• Adjoint, and evaluation of inverse of a square matrix using determinants and, test of consistency and solution of simultaneous linear equations in two or three variables using matrices

Paper 2B (B.Planning.)-Mathematics: Unit 04

• The fundamental principle of counting, permutation as an arrangement and combination as section, meaning of P (n,r) and C (n,r), simple applications

Paper 2B (B.Planning.)-Mathematics: Unit 05

• Binomial theorem for a positive integral index, general term and middle term, and simple applications

Paper 2B (B.Planning.)-Mathematics: Unit 06

• Arithmetic and geometric progressions, insertion of arithmetic, geometric means between two given numbers, relation between A.M and G.M

Paper 2B (B.Planning.)-Mathematics: Unit 07

• Real-valued functions, algebra of functions, polynomials, rational, trigonometric, logarithmic, and exponential functions, inverse function
• Graphs of simple functions
• Limits, continuity, and differentiability
• Differentiation of the sum, difference, product, and quotient of two functions
• Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite, and implicit functions, derivatives of order up to two, applications of derivatives: Rate of change of quantities, monotonic-increasing and decreasing functions
• Maxima and minima of functions of one variable

Paper 2B (B.Planning.)-Mathematics: Unit 08

• Integral as an antiderivative, fundamental integral involving algebraic, trigonometric, exponential, and logarithmic functions
• Integrations by substitution, by parts, and by partial functions, integration using trigonometric identities
• Evaluation of simple integrals of the type: ∫dx/ x² + a², ∫dx/ √x² ± a², ∫dx/ √a² - x², ∫dx/ √a² - x², ∫dx/ ax² + bx + c, ∫dx/√ax² ± bx + c, ∫(px + q) dx/ ax² + bx + c, ∫(px + q) dx/ √ax² + bx + c, ∫√a² + x² dx, ∫√x² - a² dx
• The fundamental theorem of calculus, properties of definite integrals, evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form

Paper 2B (B.Planning.)-Mathematics: Unit 09

• Ordinary differential equations, their order, and degree, the solution of differential equation by the method of separation of variables, solution of a homogeneous and linear differential equation of the type - ??/?? + ?(?)? = ?(?)

Paper 2B (B.Planning.)-Mathematics: Unit 10

• Cartesian system of rectangular coordinates in a plane, distance formula, sections formula, locus, and its equation, the slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axis
• Straight line: Various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines, the distance of a point form a line, coordinate of the centroid, orthocentre, and circumcentre of a triangle
• Circle, conic sections: A standard form of equations of a circle, the general form of the equation of a circle, its radius and central, equation of a circle when the endpoints of a diameter are given
• Circle, conic sections: Points of intersection of a line and a circle with the centre at the origin and sections of conics, equations of conic sections (parabola, ellipse, and hyperbola) in standard forms

Paper 2B (B.Planning.)-Mathematics: Unit 11

• Coordinates of a point in space, the distance between two points, section formula, directions ratios, and direction cosines, and the angle between two intersecting lines
• Skew lines, the shortest distance between them, and its equation
• Equations of a line

Paper 2B (B.Planning.)-Mathematics: Unit 12

• Vectors and scalars, the addition of vectors, components of a vector in two dimensions and three-dimensional space, scalar and vector products

Paper 2B (B.Planning.)-Mathematics: Unit 13

• Measures of discretion; Calculation of mean, median, mode of grouped and ungrouped data calculation of standard deviation, variance, and mean deviation for grouped and ungrouped data
• Probability: Probability of an event, addition and multiplication theorems of probability, Bayes theorem, probability distribution of a random variate

Paper 2B (B.Planning.)-Mathematics: Unit 14

• Trigonometrical identities and trigonometrical functions, inverse trigonometrical functions, and their properties

Paper 2B (B.Planning.)-Aptitude test: Unit 01

• Objects, texture related to architecture and build-environment, visualizing three dimensional objects from two-dimensional drawings
• Visualizing
• Different sides of three-dimensional objects
• Analytical reasoning mental ability (visual, numerical, and verbal)

Paper 2B (B.Planning.)-Aptitude test: Unit 02

• Understanding and appreciation of scale and proportions of objects, building forms and elements, colour texture harmony and contrast design and drawing of geometrical or abstract shapes and patterns in pencil
• Transformation of forms both 2D and 3D union, subtraction rotation, development of surfaces and volumes, generation of plans, elevations, and 3D views of objects, creating two-dimensional and three-dimensional compositions using given shapes and forms

Paper 2B (B.Planning.)-Planning: Unit 01

• General knowledge questions and knowledge about prominent cities, development issues, government programs, etc

Paper 2B (B.Planning.)-Planning: Unit 02

• The idea of nationalism, nationalism in India, pre-modern world, 19th century global economy, colonialism, and colonial cities, industrialization, resources, and development, types of resources, agriculture, water, mineral resources, industries
• National economy; human settlements
• Power-sharing, federalism, political parties, democracy, the constitution of India
• Economic development-economic sectors, globalization, the concept of development, poverty, population structure, social exclusion, and inequality, urbanization, rural development, colonial cities

Paper 2B (B.Planning.)-Planning: Unit 03

• Comprehension (unseen passage); map reading skills, scale, distance, direction, area, etc.; critical reasoning, understanding of charts, graphs, and tables, basic concepts of statistics and quantitative reasoning