GATE IN key terms

A scalar by which an eigenvector is scaled when a linear transformation is applied to it. Found by solving the characteristic equation det(A − λI) = 0 for the matrix A.

A nonzero vector whose direction stays unchanged when a given linear transformation acts on it, though its length may be scaled by the corresponding eigenvalue.

The number of linearly independent rows or columns in a matrix, equal to the dimension of its column space. It determines whether a linear system has a unique solution.

A scalar computed from a square matrix that indicates whether the matrix is invertible; a zero determinant means the rows are linearly dependent and no unique inverse exists.

An expansion of a smooth function as an infinite sum of terms calculated from its derivatives at a single point, used to approximate functions and evaluate limits.

The derivative of a multivariable function with respect to one variable while all other variables are held constant, measuring sensitivity along a single coordinate direction.

A vector collecting all first-order partial derivatives of a scalar function; it points in the direction of steepest increase and is central to optimisation and gradient descent.

States that for a function continuous on a closed interval and differentiable inside it, some interior point has a derivative equal to the average rate of change over the interval.

A mathematical operation that decomposes a signal into its constituent frequencies, converting a time-domain function into a frequency-domain representation used in signal processing and communications.

An integral transform that converts a function of time into a function of a complex variable, widely used to solve linear differential equations and analyse control systems.

The ratio of the Laplace transform of a system's output to that of its input under zero initial conditions, characterising the system's input-output behaviour.

A pair of graphs showing a system's magnitude and phase response versus frequency on logarithmic axes, used to assess stability margins of feedback systems.

A graphical stability test that determines closed-loop stability from the open-loop frequency response plotted in the complex plane, counting encirclements of the critical point.

The sampling theorem stating that a signal must be sampled at more than twice its highest frequency component to be reconstructed without aliasing.

A reversible electrical breakdown in heavily doped p-n junctions where a strong electric field enables tunnelling, producing a stable reference voltage in regulator diodes.

The zone around a p-n junction depleted of mobile charge carriers, leaving fixed ionised dopants that create a built-in electric field and potential barrier.

A comparison of two transistor families: the bipolar junction transistor is current-controlled and offers high transconductance, while the MOSFET is voltage-controlled with very high input impedance.

The condition in a negative-feedback op-amp circuit where the differential input voltage is approximately zero, so both input terminals sit at nearly the same potential.

A technique that maps keys to array positions using a hash function, giving average constant-time search, insert and delete operations when collisions are handled well.

A binary tree where each node's left subtree holds smaller keys and its right subtree holds larger keys, enabling ordered search in time proportional to tree height.

A description of how an algorithm's running time or memory grows with input size, expressed in Big-O, Big-Omega or Big-Theta notation while ignoring constant factors.

A state in which a set of processes each wait indefinitely for a resource held by another in the set, so none can proceed without external intervention.

A synchronisation primitive consisting of an integer and a wait queue, used to control access to shared resources and coordinate producer-consumer style interactions among processes.

A memory management scheme giving each process an independent logical address space mapped to physical frames, allowing programs larger than RAM and process isolation.

The process of organising relational schemas to reduce redundancy and update anomalies by decomposing tables according to functional dependencies and normal forms.

A constraint stating that in a relation, the value of one attribute set uniquely determines the value of another, forming the basis for key identification and normalisation.

An operation combining rows from two or more tables based on a related column, with variants such as inner, left outer and natural joins producing different result sets.

The internal moment that resists rotation at a section of a beam, caused by transverse loads and calculated from shear force distribution along the member.

The internal transverse force at a beam section equal to the algebraic sum of vertical loads on one side of that section, varying along the span.

A geometric property of a cross-section measuring its resistance to bending about an axis, computed as the integral of area times squared distance from that axis.

A field test for fresh concrete workability in which a standard cone is filled and lifted, and the vertical drop of the concrete is measured.

The mass ratio of water to cement in a concrete mix; lower values increase strength and durability, while higher values improve workability.

A dimensionless parameter comparing inertial to viscous forces in fluid flow, used to predict whether flow is laminar or turbulent.

An energy-conservation relation for steady incompressible frictionless flow stating that pressure, kinetic and potential energy terms sum to a constant along a streamline.

The thin region adjacent to a solid surface where fluid velocity rises from zero at the wall to the free-stream value, dominating viscous effects and skin friction.

The maximum possible efficiency of a heat engine operating between two reservoirs, equal to one minus the ratio of absolute sink temperature to source temperature.

A thermodynamic state property measuring energy dispersal and irreversibility; it increases for isolated systems in spontaneous processes and never decreases overall.

A principle stating that heat flows spontaneously from hot to cold bodies, that complete conversion of heat to work is impossible, and that total entropy of an isolated system cannot decrease.

The fundamental constant relating a photon's energy to its frequency, central to quantum mechanics and used to compute photon energy in the photoelectric effect.

The minimum energy needed to remove an electron from a metal surface, appearing as the threshold term in the photoelectric equation.

The wavelength associated with a moving particle, equal to Planck's constant divided by its momentum, illustrating wave-particle duality.