GATE 2025 Syllabus: A Detailed Analysis

Linear Algebra

Calculus

Vector Calculus

Complex variables

Functions of Single Variable: Limit, indeterminate forms and L’Hospital’s rule; Continuity and differentiability; Mean value theorems; Maxima and minima; Taylor’s theorem; Fundamental theorem and mean value theorem of integral calculus; Evaluation of definite and improper integrals; Applications of definite integrals to evaluate areas and volumes (rotation of a curve about an axis).

Functions of Two Variables: Limit, continuity and partial derivatives; Directional derivative; Total derivative; Maxima, minima and saddle points; Method of Lagrange multipliers; Double integrals and their applications.

Sequences and Series: Convergence of sequences and series; Tests of convergence of series with non-negative terms (ratio, root and integral tests); Power series; Taylor’s series; Fourier Series of functions of period 2π.

Gradient, divergence and curl; Line integrals and Green’s theorem.

Complex numbers, Argand plane and polar representation of complex numbers; De Moivre’s theorem; Analytic functions; Cauchy-Riemann equations.

 Ordinary Differential Equation (ODE)

Classification of second order linear partial differential equations; Method of separation of variables: One dimensional heat equation and two dimensional Laplace equation.

Working stress and Limit state design concepts; Design of tension and compression members, beams and beam- columns, column bases; Connections – simple and eccentric, beam-column connections, plate girders and trusses; Concept of plastic analysis -beams and frames.

Three-phase system and phase relationships, index properties; Unified and Indian standard soil classification system; Permeability – one dimensional flow,

Seepage through soils – two-dimensional flow, flow nets, uplift pressure, piping, capillarity, seepage force,

Principle of effective stress and quicksand condition,

Compaction of soils, One- dimensional consolidation, time rate of consolidation; Shear Strength, Mohr’s circle, effective and total shear strength parameters, Stress-strain characteristics of clays and sand; Stress Paths.

Sub-surface investigations – drilling bore holes, sampling, plate load test, standard penetration and cone penetration tests;

Earth pressure theories – Rankine and Coulomb;

Stability of slopes – finite and infinite slopes, Bishop’s method;

Stress distribution in soils – Boussinesq’s and Pressure bulbs;

Shallow foundations – Terzaghi’s and Meyerhoff’s bearing capacity theories, effect of water table;

Combined footing and raft foundation;

Contact pressure; Settlement analysis in sands and clays;

Deep foundations – dynamic and static formulae, axial load capacity of piles in sands and clays, pile load test, pile under lateral loading, pile group efficiency, and negative skin friction.

Properties of fluids, fluid statics;

Continuity, momentum, and energy equations and their applications;

Potential flow, laminar and turbulent flow,

Flow in pipes, pipe networks;

Concept of lift and drag.

Forces on immersed bodies;

Flow measurement in channels and pipes;

Dimensional analysis and hydraulic similitude;

Channel Hydraulics – Energy-depth relationships, specific energy, critical flow, hydraulic jump, uniform flow and gradually varied flow and water surface profiles.

Hydrologic cycle, precipitation, evaporation, evapo-transpiration, watershed, infiltration, unit hydrographs, hydrograph analysis, flood estimation and routing, reservoir capacity, reservoir and channel routing, surface run-off models, ground water hydrology – steady state well hydraulics and aquifers;

Application of Darcy’s law.

Types of irrigation systems and methods, Crop water requirements – Duty, delta, evapo-transpiration, Gravity dams and spillways; Lined and Unlined canals; Design of weirs on permeable foundation; Cross drainage structures.

Basics of water quality standards – physical, chemical and biological parameters, Water Quality Index; Unit processes and operations; Water requirements; Water distribution system, Drinking water treatment.

Sewerage system design, quality of domestic waste water, primary and secondary treatment, Effluent discharge standards, Sludge disposal; Reuse of treated sewage for different applications.

Geometric design of highways – cross-sectional elements, sight distances, horizontal and vertical alignments;

Geometric design of railway track – Speed and Cant;

Concept of airport runway length, calculations and corrections, taxiway and exit taxiway design.

Highway materials – desirable properties and tests;

Desirable properties of bituminous paving mixes, Design factors for flexible and rigid pavements, Design of flexible and rigid pavement using IRC codes.

Errors and their adjustment;

Traversing and triangulation survey; Total station; Horizontal and vertical curves.

Maps – scale, coordinate system; Distance and angle measurement – Levelling and trigonometric leveling

Scale, flying height; Basics of remote sensing; Geographical information system (GIS)

Functions of single variable, limit, continuity and differentiability, mean value theorems, indeterminate forms; evaluation of definite and improper integrals;

Double and triple integrals; partial derivatives, total derivative, Taylor series (in one and two variables), maxima and minima, Fourier series; gradient, divergence and curl, vector identities, directional derivatives, line, surface and volume integrals, applications of Gauss, Stokes and Green’s theorems.

First order equations (linear and nonlinear); Higher order linear differential equations with constant coefficients; Euler-Cauchy equation; initial and boundary value problems; Laplace transforms; Solutions of heat, wave and Laplace’s equations

Analytic functions; Cauchy-Riemann equations; Cauchy’s integral theorem and integral formula; Taylor and Laurent series.

Definitions of probability, sampling theorems, conditional probability; Mean, median, mode and standard deviation; Random variables, binomial, Poisson and normal distributions.

Numerical solutions of linear and non-linear algebraic equations; Integration by trapezoidal and Simpson’s rules; Single and multi-step methods for differential equations.

Free-body diagrams and equilibrium; friction and its applications including rolling friction, belt-pulley, brakes, clutches, screw jack, wedge, vehicles; trusses and frames;

Virtual work; kinematics and dynamics of rigid bodies in plane motion;

Impulse and momentum (linear and angular) and energy formulations, Lagrange’s equation.

Stress and strain, elastic constants, Poisson’s ratio; Mohr’s circle for plane stress and plane strain; thin cylinders;

Shear force and bending moment diagrams; bending shear stresses, concept of shear centre, deflection of beams

Torsion of circular shafts

Euler’s theory of columns

Energy methods; thermal stresses; strain gauges and rosettes; testing of materials with universal testing machine;

Testing of hardness and impact strength.

Fluid properties; fluid statics, forces on submerged bodies, stability of floating bodies;

Differential equations of continuity and momentum; Bernoulli’s equation; dimensional analysis;

Viscous flow of incompressible fluids, boundary layer, elementary turbulent flow, flow through pipes, head losses in pipes, bends and fittings;

Basics of compressible fluid flow

Modes of heat transfer; one dimensional heat conduction, resistance concept and electrical analogy, heat transfer through fins;

unsteady heat conduction, lumped parameter system, Heisler’s charts;

Thermal boundary layer, dimensionless parameters in free and forced convective heat transfer; heat transfer correlations for flow over flat plates and through pipes, effect of turbulence;

heat exchanger performance, LMTD and NTU methods;

Radiative heat transfer, Stefan- Boltzmann law, Wien’s displacement law, black and grey surfaces, view factors, radiation network analysis.

Thermodynamic systems and processes; properties of pure substances, behavior of ideal and real gases;

Zeroth and first laws of thermodynamics, calculation of work and heat in various processes;

Second law of thermodynamics;

Thermodynamic property charts and tables, availability and irreversibility; thermodynamic relations.

Power Engineering: Air and gas compressors; vapour and gas power cycles, concepts of regeneration and reheat.

I.C. Engines: Air-standard Otto, Diesel, and dual cycles.

Turbo machinery: Impulse and reaction principles, velocity diagrams, Pelton-wheel, Francis and Kaplan turbines.

Different types of castings, design of patterns, moulds and cores; solidification and cooling; riser and gating design.

Principles of welding, brazing, soldering and adhesive bonding

Plastic deformation and yield criteria; fundamentals of hot and cold working processes; load estimation for bulk (forging, rolling, extrusion, drawing) and sheet (shearing, deep drawing, bending) metal forming processes; principles of powder metallurgy.

Principles of welding, brazing, soldering, and adhesive bonding.

Mechanics of machining; basic machine tools;

Principles of non-traditional machining processes; principles of work holding, jigs and fixtures.

Single and multi-point cutting tools, tool geometry and materials, tool life and wear; economics of machining.

NC/CNC machines and CNC programming.

Limits, fits and tolerances; linear and angular measurements; comparators;

Alignment and testing methods; tolerance analysis in manufacturing and assembly

Interferometry; form and finish measurement;

Concepts of coordinate-measuring machine (CMM).

Basic concepts of CAD/CAM and their integration tools;

Additive manufacturing

Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial Derivatives,

Maxima and minima, Multiple Integrals, Fourier series, Vector identities, Directional derivatives, Line integral, Surface integral, Volume integral, Stokes’s theorem, Gauss’s theorem, Divergence theorem and Green’s theorem.

Section 2: Electric Circuits

Network elements: Idea voltage and current sources, R,L, C, M elements.

Network solution methods: KCL, KVL, Node and Mesh Analysis;

Network Theorems: Thevenin’s, Norton’s, Superposition and Maximum Power Transder theorem;

Transient response of dc and ac networks, sinusoidal steady-state analysis, resonance, two port networks, balanced three-phase circuits, star-delta transformation, complex power, and power-factor in ac circuits.

Section 3: Electromagnetic Fields

Coulomb’s Law, Electric Field Intensity, Electric Flux Density, Gauss’s Law, Divergence, Electric field and potential due to point, line, plane and spherical charge distributions;

Effect of the dielectric medium, Capacitance of simple configurations,

Biot‐Savart’s law, Ampere’s law, Curl,

Faraday’s law, Lorentz force, Inductance, Magnetomotive force, Reluctance, Magnetic circuits,

Self and Mutual inductance of simple configurations.

Section 4: Signals and Systems

Representation of continuous and discrete‐time signals, shifting and scaling operations,

Fourier series representation of continuous periodic signals,

Laplace Transform and z-Transform,

Linear Time Invariant and Causal systems,

Sampling theorem, Applications of Fourier Transform;

R.M.S. value, average value calculation for any general periodic waveform.

Section 5: Electrical Machines

Single phase transformer: equivalent circuit, phasor diagram, open circuit and short circuit tests, regulation and efficiency;

Three phase transformers: connections, vector groups, parallel operation;

Three phase induction motors: principle of operation, types, performance, torque-speed characteristics, no-load and blocked rotor tests, equivalent circuit, starting and speed control;

Operating principle of single phase induction motors;

Synchronous machines: cylindrical and salient pole machines, performance, regulation and parallel operation of generators, starting of synchronous motor, characteristics;

Auto‐transformer, Electromechanical energy conversion principles;

DC machines: separately excited, series and shunt, motoring and generating mode of operation and their characteristics, starting and speed control of dc motors;

Types of losses and efficiency calculations of electric machines.

Section 6: Power Systems

Basic concepts of electrical power generation, ac and dc transmission concepts,

Models and performance of transmission lines and cables, Series and shunt compensation, Economic load dispatch (with and without considering transmission losses), Electric field distribution and insulators;

Distribution systems, Per‐unit quantities, Bus admittance matrix, Gauss- Seidel and Newton-Raphson load flow methods;

Principles of over‐current, differential and distance protection;

Circuit breakers, System stability concepts, Equal area criterion.

Voltage and Frequency control, Power factor correction, Symmetrical components, Symmetrical and unsymmetrical fault analysis.

Mathematical modelling and representation of systems, Feedback principle, transfer function;

Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci, Stability analysis, Lag, Lead and Lead‐Lag compensators;

P, PI and PID controllers; State space model, State transition matrix.

Block diagrams and Signal flow graphs, Transient and Steady‐state analysis of linear time invariant systems.

Section 8: Electrical and Electronic Measurements

Bridges and Potentiometers, Measurement of voltage, current, power, energy and power factor; Instrument transformers,

Digital voltmeters and multi-meters, Phase, Time and Frequency measurement;

Oscilloscopes, Error analysis.

Section 9: Analog and Digital Electronics

Simple diode circuits: clipping, clamping, rectifiers;

Amplifiers: Biasing, Equivalent circuit and Frequency response; Oscillators and Feedback amplifiers;

Operational amplifiers: Characteristics and applications;

Single stage active filters

Active filters: Sallen Key, Butterworth, VCOs and timers, Combinational and Sequential logic circuits, Multiplexers, Demultiplexers, Schmitt trigger, Sample and hold circuits, A/D and D/A converters.

Section 10: Power Electronics

Stage V-I characteristics and firing/gating circuits  for Thyristor, MOSFET, IGBT;

DC to DC conversion: Buck, Boost and Buck-Boost converters; Single and three phase configuration of uncontrolled rectifiers;

Voltage and Current commutated Thyristor based convertors;

Bidirectional ac to dc voltage source converters; Power factor and Distortion factor of ac to dc converters, Single phase and three phase voltage and current source inverters, Sinusoidal pulse width modulation.

Node and mesh analysis;

Superposition, Thevenin and Norton’s theorem, maximum power transfer;

Time and frequency domain analysis of linear circuits: RL, RC and RLC circuits; solution of network equations using Laplace transform;

Linear 2‐port network parameters and Wye-Delta transformation.

Discrete-time signals: discrete-time Fourier transform (DTFT), DFT, FFT, Z-transform, interpolation of discrete-time signals;

LTI systems: definition and properties, causality, stability, impulse response, convolution, poles and zeros, parallel and cascade structure, frequency response, group delay, phase delay, digital filter design techniques.

Number representations: binary, integer, and floating point numbers; Combinatorial circuits: Boolean algebra, minimization of functions using Boolean identities and Karnaugh map, logic gates and their static CMOS implementations, arithmetic circuits, code converters, multiplexers, decoders;

Sequential circuits: latches and flip‐flops, counters, shift‐registers and finite state machines;

Data converters: sample and hold circuits, ADCs and DACs; Semiconductor memories: ROM, SRAM, DRAM;

Computer organization: Machine instructions and addressing modes, ALU, data-path and control unit, and instruction pipelining.

Random processes: auto correlation and power spectral density, properties of white noise, filtering of random signals through LTI systems;

Information theory: entropy, mutual information and channel capacity theorem;

Digital communications: PCM, DPCM, digital modulation schemes, amplitude, phase and frequency shift keying (ASK, PSK, FSK, QAM), MAP and ML decoding, matched filter receiver, calculation of bandwidth, SNR and BER;

Fundamentals of error correction, Hamming codes and CRC;

Timing and frequency synchronization, inter-symbol interference and its mitigation;

Basics of TDMA, FDMA and CDMA.

Analog communications: amplitude modulation and demodulation, angle modulation and demodulation, spectra of AM and FM, super heterodyne receivers.

Maxwell’s equations: differential and integral forms and their interpretation, boundary conditions, wave equation, and Poynting vector;

Plane waves and properties: reflection and refraction, polarization, phase and group velocity, propagation through various media, skin depth;

Transmission lines: equations, characteristic impedance, impedance matching, impedance transformation, S-parameters, Smith chart;

Rectangular and circular waveguides; light propagation in optical fibers, dipole and monopole antennas, and linear antenna arrays.

Concept of layering: OSI and TCP/IP Protocol Stacks; Basics of packet, circuit and virtual circuit switching;

Data link layer: framing, error detection, Medium Access Control, Ethernet Bridging;

Routing protocols: shortest path, flooding, distance vector and link state routing; Fragmentation and IP addressing; IPv4, CIDR notation, Basics of IP support protocols (ARP, DHCP, ICMP), Network Address Translation (NAT); Transport layer: flow control and congestion control, UDP, TCP, sockets; Application layer protocols: DNS, SMTP, HTTP, FTP, and Email.