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Matrices around order 3

Definition of matrix and its order. Different types of matrices (rectangular, square, row, column, upper triangular, lower triangular, diagonal, scalar, identity, null). Equality of two matrices, addition, subtraction, multiplication of a matrix by a scalar, and multiplication of two matrices. Transpose of a matrix, symmetric and skew-symmetric matrices, simple problems. Singular and non-singular matrices, adjoint and inverse of a matrix of order 3. Eigen Values of matrix up to order 3. Statement of Cauchy-Hamilton Theorem and application for determining the inverse of a matrix. Diagonalization of matrices.

Determinant around order 3

Definition and expansion of determinants of order 2 and 3. Minor and cofactors. Elementary properties of Determinants (statements only). Solutions of linear simultaneous equations (up to 3 unknowns) by Cramer’s Rule. The rank of a matrix is up to order 3. Linear homogeneous & non-homogeneous systems of equations – statements of the relevant results.

Complex Number

Definition of complex numbers, Cartesian and polar. The exponential form of complex numbers. Modulus, amplitude, and conjugate of a complex number. Algebra of complex numbers (Equality, Addition, Subtraction, Multiplication). Cube roots of unity and their properties. De Moivre’s theorem (statements only).

Co-ordinate Geometry(2D)

Concept of polar coordinates and its relation to Cartesian coordinates. The conic section in 2D – Definition, simple properties, Tangents and Normal.

Vector Algebra

Definition of a vector quantity. Concept of Position vector and Ratio formula. Rectangular resolution of a vector. Algebra of vectors – equality, addition, subtraction, and scalar multiplication. Scalar (Dot) product of two vectors with properties. Vector (cross) product of two vectors with properties. Applications: The application of dot product in work done by a force, the projection of one vector upon another, and the application of cross product in finding vector area and moment of a force. Scalar and vector triple product and their geometrical interpretations. Linear combination of 3 vectors. Linear dependence and independence of vectors.

Differential Calculus

Concept of function of one variable – Domain and range. Type of different functions, including periodic functions. Limit and continuity. Types of discontinuity. Derivative of functions (1st order and 2nd order). Statements & Applications of Rolls & Mean Value Theorem. Indeterminant Form. L’Hospital’s rule.

Integral Calculus

Definition of Integration as inverse process of differentiation. Rules for integration (sum, difference, scalar multiple). Integration of standard functions. Integration by substitution. Integration by parts. Integration by partial fraction. Definition of definite integral & properties of definite integral. Application of definite integral – area of bounded region.

Partial Differentiation & Application of Derivative

Definition and meaning of partial derivative. Evaluation of partial derivatives. Definition and examples of homogeneous functions. Euler’s theorem (1st order) on homogeneous functions for 2 variables (without proof). Geometric meaning of derivative. Rate measurement. Maxima and Minima (one variable)

Probability

Definition of random experiment, sample space, event, occurrence & types of events (e.g., Impossible, mutually exclusive, Exhaustive, equally likely). Classical definition of probability, simple problems. Statements of total probability, compound probability & Base Theorem.

Units, Dimensions, and Measurement

CGS, MKS, SI Dimensions of common physical quantities, dimensional analysis. Common measurement errors. Error, accuracy, precision, resolution, significant figure.

Kinematics

Speed, velocity, acceleration, uniform/non-uniform, rectilinear/ circular. Position/ velocity-time graph. Resolution and composition of vectors, scalar multiplication of vectors.

Laws of Motion

Newton’s laws of motion. Force, momentum, inertia, moment of inertia, impulse, couples, moment. Conditions of equilibrium. Conservation of momentum. Centripetal and centrifugal forces. Angular displacement/ velocity/ acceleration/ momentum, torque. Static and dynamic friction, angle of repose, banking of roads.

Work, Power, Energy & Gravitation

Definition, measures, and units. Law of conservation of energy. Kinetic and potential. The universal law of Acceleration due to gravity and its variation on/ above/ below Earth’s surface. Gravitational potential energy. Vertical linear/ vertical circular/ projectile motion.

Elasticity

Deforming force and restoring force, elastic, and plastic body. Stress-strain relationship, Hook’s law, Young’s modulus, Bulk modulus, Rigidity modulus, Poisson’s ratio, and relation between them. Elastic energy.

Thermodynamics

Thermal equilibrium and the Zeroth law of thermodynamics. Heat, work, temperature, and internal. The first law of thermodynamics. Specific heats of gas, their relation, and their ratio. Isothermal, isobaric, isochoric, and adiabatic processes.

Fluid Mechanics or Hydrostatics

Pascal’s law. Hydraulic lift. Buoyancy. Conditions of equilibrium of a floating body. Archimedes’ principle. Streamline flow and turbulent flow of a fluid, critical velocity. Equation of continuity and Bernoulli’s theorem. Viscosity, Newton’s formula for viscous force, coefficient of viscosity. Stokes law and terminal velocity. Effect of temperature on viscosity.

Reflection & Refraction of Light

Reflection of light in a plane & through a plane surface. Laws of refraction. Refractive index & its relationship with the velocity of light in different media. Total internal reflection, critical angle, and the principle of optical fibre.

Lens & Photoelectricity

Convex and concave lenses. Formation of image. Relation between u, v, f. Power of a lens (in different mediums). Equivalent focal length & power of two thin lenses in contact. Photoemission, Work function. Photoelectric current, its variation with intensity and frequency of incident radiation. Stopping potential, Threshold frequency. Concept of photon. Einstein’s photoelectric principle of solar photovoltaic cells.

Atomic Structure

Bohr model of atom, de Broglie wave equation, Quantum numbers, Orbits and Orbitals, Aufbau principle, Pauli’s Exclusion principle, Hund’s rule of maximum multiplicity, electronic configuration of elements. Definition of Atomic number, Mass number, isotopes, isotones, and isobars. Concept of hybridization sp3, sp2, sp, and shape of molecules.

Chemical Bonding

Electrovalent, Covalent, and coordinate bonds, H-bond in HF, water, and ice. Classification of solids – crystalline and amorphous. Relationship between structure and properties of crystalline solids such as ionic, covalent, and molecular solids. Covalent bonds in Carbon, Silicon, and Germanium.

Acids, Bases & Salts

Avogadro number, Mole concept, weight and volume relations. Acids, Bases and Salts (Arrhenius and Lewis concept). Acidity, basicity, neutralization reaction, hydrolysis of Salts. Equivalent Weight of acids, bases, & salts; strength of solution- normality, molarity, molality, formality and percentage strength, standard solution- primary and secondary standards, concept of pH, and pH scale. Indicators and choice of indicator, principles of acidimetry and alkalimetry. Buffer Solubility product principle. Common ion effect with relation to group analysis.

Oxidation, Reduction, Electrochemistry

Oxidation and Reduction by electronic concept, balancing chemical equations by ion-electron method, Redox Titration, Electrolysis, Arrhenius theory, Faraday’s Laws, electrolysis of CuSO4 solution using Pt-electrode and Cu-electrode. Application of electrolysis such as Electroplating, Electro-refining and Electrotyping, Electrochemical Cells, Primary Cell- Dry Cell, Secondary Cell- Lead storage cell, Electrochemical series.

Chemical Equilibrium

Reversible and irreversible reactions, exothermic and endothermic reactions, chemical equilibrium, Le Chatelier’s Industrial preparation of Ammonia by Haber’s Process, Nitric acid by Ostwald’s process, and Sulphuric acid by Contact Process. Catalyst and catalysis.

Metallurgy

Minerals, Ores, Gangue, Flux, Slag, General method of extraction of metals concerning Iron, Copper, and Aluminium. Definition of Alloy, purposes of making Alloy, Composition and uses of alloys such as brass, bronze, German silver, duralumin, nichrome, bell metal, gun metal, Monel metal, alnico, Dutch metal, babbitt metal, stainless steel. Amalgams, properties and uses of cast iron, wrought iron, steel and sponge iron. Uses of different alloy steels.

Water

Soft and Hard water, action of soap on water, types of Hardness, causes of hardness, units of hardness, cons of using hard water, estimation of total hardness by EDTA method, removal of hardness by Permutit process, Ion-exchange process, phosphate conditioning, and Calgon treatment.

Organic Chemistry

Organic compounds, Homologous series, Functional groups, Isomerism, Properties and preparation of Methane, Ethylene and Acetylene, Methylated spirit, Rectified spirit, Power alcohol, Proof spirit. Uses of Benzene, Naphthalene, and Phenol.

Electrical components, measuring instruments

Electrical components, measuring instruments: Resistors, capacitors, inductors. Principle of operation of PMMC & MI type Instrument. Principles of Voltage and Current Wheatstone bridge.

Transformers

Basic principle and construction. BH characteristics. Transformation ratio. Equivalent circuit, losses, regulation, Auto transformer.

Electrical machines

Types of motors and their applications. Generation of rotating magnetic fields. Basic construction and working of DC motor, single-phase induction. Torque-Speed characteristics. Speed control.

DC sources and circuits

Voltage and current sources, their transformation, Thevenin’s, Norton’s, Kirchhoff current and voltage laws. Superposition, equivalence. Determination of branch current/ voltage in passive circuits. Time domain characteristics of reactive components.

AC sources and circuits

Amplitude, frequency, time-period, phase, peak/average/ RMS. Exponential, complex, and phasor form of representation of sinusoidal voltage and current. Impedance, power factor, active/reactive power. Voltage and current relationship in star-delta connections. Behaviour of simple RLC circuits with AC excitation.

Basic semiconductor devices

Energy level diagrams of insulator, conductor & semiconductor. Intrinsic & extrinsic, P-type, N-type semiconductors. P-N junction diode. PNP and NPN transistors, MOSFET, CMOS.

Analog Circuits

OPAMP, virtual ground, inverting and non-inverting amplifiers, adder, subtractor, integrator, differentiator.

Digital Electronics

Boolean Algebra, basic gates (AND, OR, NOT, NAND, NOR, Ex-OR, Ex-NOR) – truth tables, symbols, and combination of gates.