Basic Engineering Math interview questions

150 questions with answers (MAT-001 to MAT-150). Try each one aloud before reading the answer.

Linear Algebra

MAT-001 · What is the rank of a matrix?

The maximum number of linearly independent rows (or, equivalently, columns). It equals the number of non-zero rows in the row-echelon form.

MAT-002 · What do eigenvalues and eigenvectors mean?

For a square matrix A, Av = λv. The eigenvector v keeps its direction under the transformation A and is only scaled by the eigenvalue λ.

MAT-003 · When is a matrix invertible?

A square matrix is invertible when its determinant ≠ 0, which is the same as having full rank, linearly independent columns and no zero eigenvalue.

MAT-004 · What is a symmetric matrix?

A matrix with A = Aᵀ. Real symmetric matrices have real eigenvalues and orthogonal eigenvectors, so they can always be diagonalized by an orthogonal matrix.

MAT-005 · What is an orthogonal matrix?

A real square matrix Q with QᵀQ = I, so Q⁻¹ = Qᵀ. Its columns are orthonormal, det Q = ±1, and it preserves lengths and angles (rotations and reflections).

MAT-006 · What is the trace of a matrix?

The sum of the diagonal elements. It equals the sum of the eigenvalues and satisfies tr(AB) = tr(BA).

MAT-007 · What does the determinant represent geometrically?

The signed factor by which the linear map scales area (2D) or volume (3D). A negative sign means orientation is reversed; zero means the space is flattened. It also equals the product of the eigenvalues.

MAT-008 · What is the difference between homogeneous and non-homogeneous linear systems?

A homogeneous system is Ax = 0 and always has the trivial solution x = 0. A non-homogeneous system Ax = b (b ≠ 0) may have no, one or infinitely many solutions.

MAT-009 · What is linear independence?

Vectors v₁…vₙ are linearly independent if c₁v₁ + … + cₙvₙ = 0 only when all cᵢ = 0. No vector in the set can be written as a combination of the others.

MAT-010 · What is a skew-symmetric matrix?

A matrix with Aᵀ = −A. Its diagonal entries are zero and its real eigenvalues are zero; all other eigenvalues are purely imaginary.

MAT-011 · What is the null space of a matrix?

The set of all vectors x with Ax = 0. Its dimension is the nullity, and rank + nullity = number of columns (rank–nullity theorem).

MAT-012 · What are a basis and the dimension of a vector space?

A basis is a set of linearly independent vectors that spans the space. The dimension is the number of vectors in any basis, e.g. 3 for ℝ³.

MAT-013 · What is the characteristic equation of a matrix?

det(A − λI) = 0. It is a polynomial of degree n whose roots are the eigenvalues of the n×n matrix A.

Differential Calculus

MAT-014 · What is a derivative, physically?

The instantaneous rate of change of a quantity, geometrically the slope of the tangent to the curve. Velocity is the derivative of displacement with respect to time.

MAT-015 · What is the difference between continuity and differentiability?

Continuity means no breaks: lim f(x) = f(a). Differentiability means a unique tangent exists. Differentiability implies continuity, but a continuous function can have corners where it is not differentiable.

MAT-016 · What is a partial derivative?

The rate of change of a multivariable function with respect to one variable while all others are held constant, e.g. ∂f/∂x of f(x, y).

MAT-017 · What does the second derivative tell you?

The rate of change of the slope. f″ > 0 means concave up, f″ < 0 means concave down. Physically, the second derivative of displacement is acceleration.

MAT-018 · How do you find the local maxima and minima of f(x)?

Solve f′(x) = 0 for critical points, then apply the second-derivative test: f″ > 0 gives a minimum, f″ < 0 a maximum, and f″ = 0 is inconclusive.

MAT-019 · What is the chain rule?

The derivative of a composite function: d/dx f(g(x)) = f′(g(x))·g′(x). Example: d/dx sin(x²) = 2x cos(x²).

MAT-020 · What is a limit?

The value f(x) approaches as x approaches a point. The limit exists only if the left-hand and right-hand limits exist and are equal.

MAT-021 · What is L'Hôpital's rule?

For an indeterminate form 0/0 or ∞/∞, lim f(x)/g(x) = lim f′(x)/g′(x), provided the right-hand limit exists.

MAT-022 · What is a Taylor series?

An expansion of a smooth function about x = a: f(x) = Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n!. Truncating it gives polynomial approximations used in linearization and numerical methods.

MAT-023 · What is a Maclaurin series? Give an example.

A Taylor series about a = 0. Example: eˣ = 1 + x + x²/2! + x³/3! + …, valid for all x.

MAT-024 · State the Mean Value Theorem.

If f is continuous on [a, b] and differentiable on (a, b), there is at least one c in (a, b) with f′(c) = [f(b) − f(a)]/(b − a).

MAT-025 · State Rolle's theorem.

If f is continuous on [a, b], differentiable on (a, b) and f(a) = f(b), then f′(c) = 0 for at least one c in (a, b). It is a special case of the Mean Value Theorem.

MAT-026 · What is a total differential?

For f(x, y), df = (∂f/∂x)dx + (∂f/∂y)dy. It approximates the change in f for small changes in the variables and is used for error estimation.

Integral Calculus

MAT-027 · What does a definite integral represent?

The net signed area under f(x) between a and b, or more generally the accumulation of a quantity, such as distance from velocity or work from force.

MAT-028 · State the Fundamental Theorem of Calculus.

If F′ = f and f is continuous on [a, b], then ∫ₐᵇ f(x)dx = F(b) − F(a). Also, d/dx ∫ₐˣ f(t)dt = f(x), so differentiation and integration are inverse operations.

MAT-029 · What is the difference between definite and indefinite integrals?

An indefinite integral is a family of antiderivatives F(x) + C. A definite integral has limits and gives a number.

MAT-030 · What is integration by parts?

∫u dv = uv − ∫v du, derived from the product rule. The ILATE order (inverse trig, log, algebraic, trig, exponential) helps choose u.

MAT-031 · What is an improper integral?

An integral with an infinite limit or an integrand that is unbounded on the interval. It is defined as a limit and converges only if that limit is finite.

MAT-032 · What is the Gamma function?

Γ(n) = ∫₀^∞ xⁿ⁻¹e⁻ˣ dx for n > 0. It satisfies Γ(n + 1) = nΓ(n), so Γ(n + 1) = n! for integers, and Γ(1/2) = √π.

MAT-033 · What is the Beta function?

B(m, n) = ∫₀¹ xᵐ⁻¹(1 − x)ⁿ⁻¹ dx for m, n > 0. It relates to Gamma by B(m, n) = Γ(m)Γ(n)/Γ(m + n).

MAT-034 · What are double integrals used for?

To integrate over a 2D region: area (∬dA), volume under a surface, mass of a lamina, centroids and moments of inertia.

MAT-035 · How do odd and even functions behave when integrated over symmetric limits?

Over [−a, a], an odd function integrates to 0 and an even function gives 2∫₀ᵃ f(x)dx.

MAT-036 · What is the average value of a function on [a, b]?

f_avg = (1/(b − a))∫ₐᵇ f(x)dx. The Mean Value Theorem for integrals says a continuous f attains this value somewhere in the interval.

MAT-037 · Why does a Jacobian appear when changing variables in a multiple integral?

The area element transforms as dx dy = |J| du dv, where J = ∂(x, y)/∂(u, v). For polar coordinates |J| = r, so dx dy = r dr dθ.

MAT-038 · What is the area under y = x² from x = 0 to 1?

∫₀¹ x² dx = [x³/3]₀¹ = 1/3.

MAT-039 · How do you find the volume of a solid of revolution?

Disc method about the x-axis: V = π∫ₐᵇ y² dx. Shell method about the y-axis: V = 2π∫ₐᵇ x·y dx.

Vector Calculus

MAT-040 · What is the gradient of a scalar field?

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z). It points in the direction of steepest increase, its magnitude is that maximum rate, and it is normal to level surfaces.

MAT-041 · What is divergence?

∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z, a scalar giving the net outward flux per unit volume. Positive means a source, negative means a sink.

MAT-042 · What is curl?

∇×F, a vector measuring the local rotation or circulation per unit area of a field. Its direction is the axis of rotation by the right-hand rule.

MAT-043 · What is a solenoidal field?

A field with zero divergence, ∇·F = 0, so it has no sources or sinks. The magnetic field B and the velocity of an incompressible fluid are examples.

MAT-044 · What is an irrotational field?

A field with ∇×F = 0. On a simply connected domain it is conservative: F = ∇φ, and its line integrals are path-independent.

MAT-045 · What is the difference between the dot and cross products?

a·b = |a||b|cos θ is a scalar measuring alignment (used for work). a×b = |a||b|sin θ n̂ is a vector perpendicular to both (used for torque and area).

MAT-046 · What is a directional derivative?

The rate of change of f in the direction of a unit vector û: D_û f = ∇f·û. It is maximal along ∇f.

MAT-047 · What is the Laplacian?

∇²f = ∇·∇f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z². It measures how a value differs from its neighbourhood average and appears in heat, wave and potential equations.

MAT-048 · State Green's theorem.

For a positively oriented simple closed curve C enclosing region R: ∮_C (P dx + Q dy) = ∬_R (∂Q/∂x − ∂P/∂y) dA.

MAT-049 · State Stokes' theorem.

∮_C F·dr = ∬_S (∇×F)·dS. The circulation around a closed curve equals the flux of curl through any surface bounded by it.

MAT-050 · State the Gauss divergence theorem.

∯_S F·dS = ∭_V (∇·F) dV. The outward flux through a closed surface equals the volume integral of the divergence inside.

MAT-051 · What is the scalar triple product?

a·(b×c), which equals the determinant of the three vectors. Its magnitude is the volume of the parallelepiped they span; zero means the vectors are coplanar.

Ordinary Differential Equations

MAT-052 · What are the order and degree of an ODE?

The order is the highest derivative present. The degree is the power of that highest derivative once the equation is free of radicals and fractions in the derivatives.

MAT-053 · What is the difference between an ODE and a PDE?

An ODE has one independent variable and ordinary derivatives. A PDE has two or more independent variables and partial derivatives.

MAT-054 · What is the difference between a general and a particular solution?

A general solution contains as many arbitrary constants as the order. A particular solution fixes those constants using initial or boundary conditions.

MAT-055 · What is a separable ODE?

One that can be written dy/dx = g(x)h(y). Rearranging to dy/h(y) = g(x)dx lets you integrate both sides separately.

MAT-056 · What is an exact differential equation?

M(x, y)dx + N(x, y)dy = 0 is exact if ∂M/∂y = ∂N/∂x. Then a potential F exists with dF = 0, so the solution is F(x, y) = C.

MAT-057 · What is an integrating factor for a first-order linear ODE?

For y′ + P(x)y = Q(x), multiply by IF = e^(∫P dx). The left side becomes d(y·IF)/dx, so y·IF = ∫Q·IF dx + C.

MAT-058 · What is the auxiliary equation of a constant-coefficient ODE?

For ay″ + by′ + cy = 0, substitute y = eᵐˣ to get am² + bm + c = 0. Its roots determine the form of the solution.

MAT-059 · What are the complementary function and particular integral?

For a linear non-homogeneous ODE, y = CF + PI. The CF solves the homogeneous equation; the PI is any one solution of the full equation.

MAT-060 · What is the difference between an initial value problem and a boundary value problem?

An IVP specifies the solution and its derivatives at one point. A BVP specifies conditions at two or more points, such as both ends of a beam.

MAT-061 · What is a homogeneous linear ODE?

A linear ODE whose right-hand side (forcing term) is zero, e.g. y″ + 3y′ + 2y = 0. Any linear combination of its solutions is also a solution.

MAT-062 · What is the Wronskian?

For two solutions, W = y₁y₂′ − y₂y₁′. If W ≠ 0, the solutions are linearly independent and form a fundamental set.

MAT-063 · What is the solution of dy/dx = ky?

y = Ce^(kx). It models exponential growth (k > 0), such as populations, or decay (k < 0), such as radioactivity and RC discharge.

MAT-064 · What is a Bernoulli differential equation?

y′ + P(x)y = Q(x)yⁿ. The substitution v = y¹⁻ⁿ turns it into a linear first-order ODE in v.

Partial Differential Equations

MAT-065 · What is a partial differential equation?

An equation relating an unknown function of several variables to its partial derivatives, e.g. u(x, t) in the heat equation. PDEs model fields varying in space and time.

MAT-066 · What is the one-dimensional heat equation?

∂u/∂t = α²∂²u/∂x², where u is temperature and α² is the thermal diffusivity (m²/s). It describes diffusion.

MAT-067 · What is the one-dimensional wave equation?

∂²u/∂t² = c²∂²u/∂x², where c is the wave speed. It models vibrating strings, sound and transmission lines.

MAT-068 · What is Laplace's equation?

∇²u = 0. It governs steady-state temperature, electrostatic potential in charge-free regions and ideal potential flow.

MAT-069 · What is Poisson's equation?

∇²u = f(x, y, z), Laplace's equation with a source term. Example: ∇²V = −ρ/ε₀ in electrostatics.

MAT-070 · What is the order of a PDE?

The order of the highest partial derivative in the equation. The heat, wave and Laplace equations are all second order.

MAT-071 · When is a PDE linear?

When the unknown and its partial derivatives appear only to the first power and are not multiplied together; the coefficients may depend on the independent variables.

MAT-072 · What is the method of separation of variables?

Assume a product solution such as u(x, t) = X(x)T(t), which splits the PDE into ODEs linked by a separation constant. Solutions are then superposed to fit the conditions.

MAT-073 · What are Dirichlet and Neumann boundary conditions?

Dirichlet specifies the value of u on the boundary (e.g. fixed temperature). Neumann specifies the normal derivative ∂u/∂n (e.g. heat flux; zero means insulated).

MAT-074 · What is a harmonic function?

A twice continuously differentiable function satisfying Laplace's equation ∇²u = 0. The real and imaginary parts of an analytic function are harmonic.

MAT-075 · What is the difference between steady-state and transient solutions?

A steady-state solution does not change with time (∂u/∂t = 0). The transient part decays or evolves with time from the initial condition toward that steady state.

MAT-076 · How many initial conditions do the heat and wave equations need?

The heat equation is first order in time and needs one: u(x, 0). The wave equation is second order in time and needs two: u(x, 0) and ∂u/∂t(x, 0).

Complex Variables

MAT-077 · What is the polar form of a complex number?

z = x + iy = r(cos θ + i sin θ) = re^(iθ), where r = |z| = √(x² + y²) and θ = arg z = atan2(y, x).

MAT-078 · State Euler's formula.

e^(iθ) = cos θ + i sin θ. With θ = π it gives e^(iπ) + 1 = 0. It underpins phasor analysis of AC circuits.

MAT-079 · What is De Moivre's theorem?

(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. It is used to compute powers and roots of complex numbers and to derive multiple-angle identities.

MAT-080 · What is the complex conjugate and why is it useful?

For z = x + iy, z̄ = x − iy. Since z·z̄ = |z|², it is used to rationalise denominators and to compute magnitudes and power.

MAT-081 · What is an analytic function?

A complex function that is differentiable at every point of some neighbourhood of a point (or of a region). Analytic functions have derivatives of all orders and convergent Taylor series.

MAT-082 · What is a singularity of a complex function?

A point where the function fails to be analytic, such as z = 0 for 1/z. Isolated singularities are classified as removable, poles or essential.

MAT-083 · What is a pole?

An isolated singularity z₀ where |f(z)| → ∞ and (z − z₀)ᵐf(z) is analytic and non-zero at z₀ for some smallest integer m, the order of the pole.

MAT-084 · What is a residue?

The coefficient a₋₁ of 1/(z − z₀) in the Laurent series of f about an isolated singularity z₀. Residues are used to evaluate contour integrals.

MAT-085 · What is a Laurent series?

A series f(z) = Σ aₙ(z − z₀)ⁿ over all integers n, including negative powers. It is valid in an annulus around a singularity, where a Taylor series is not.

MAT-086 · What is an entire function?

A function analytic at every finite point of the complex plane. Polynomials, eᶻ, sin z and cos z are entire.

MAT-087 · What are the nth roots of unity?

The solutions of zⁿ = 1: z_k = e^(i2πk/n) for k = 0, 1, …, n − 1. They lie equally spaced on the unit circle and sum to zero for n > 1.

MAT-088 · State Cauchy's integral theorem.

If f is analytic inside and on a simple closed contour C (in a simply connected domain), then ∮_C f(z)dz = 0.

Probability

MAT-089 · What is the classical definition of probability?

P(A) = (number of favourable outcomes)/(total number of equally likely outcomes). It always lies between 0 and 1.

MAT-090 · What is the difference between mutually exclusive and independent events?

Mutually exclusive events cannot occur together: P(A∩B) = 0. Independent events do not affect each other: P(A∩B) = P(A)P(B).

MAT-091 · What is conditional probability?

The probability of A given that B has occurred: P(A|B) = P(A∩B)/P(B), defined for P(B) > 0.

MAT-092 · State the addition rule of probability.

P(A∪B) = P(A) + P(B) − P(A∩B). For mutually exclusive events it reduces to P(A) + P(B).

MAT-093 · What is a random variable?

A function that assigns a numerical value to each outcome of a random experiment, such as the number of heads in three tosses.

MAT-094 · What is the difference between discrete and continuous random variables?

A discrete variable takes countable values and has a probability mass function. A continuous variable takes values in an interval and has a probability density function, with probabilities given by areas.

MAT-095 · What is the expected value of a random variable?

The probability-weighted mean: E[X] = Σ x·p(x) for discrete or ∫ x·f(x)dx for continuous variables. It is linear: E[aX + b] = aE[X] + b.

MAT-096 · What is variance?

Var(X) = E[(X − μ)²] = E[X²] − (E[X])². It measures spread; its square root is the standard deviation.

MAT-097 · What is the binomial distribution?

The number of successes in n independent trials with success probability p: P(X = k) = C(n, k)pᵏ(1 − p)ⁿ⁻ᵏ. Mean np, variance np(1 − p).

MAT-098 · What is the Poisson distribution?

The number of events in a fixed interval at constant average rate λ: P(X = k) = e^(−λ)λᵏ/k!. Mean and variance both equal λ.

MAT-099 · What is the uniform distribution?

On [a, b], f(x) = 1/(b − a), so every equal sub-interval is equally likely. Mean (a + b)/2, variance (b − a)²/12.

MAT-100 · What is the exponential distribution?

f(x) = λe^(−λx) for x ≥ 0, modelling the time between Poisson events. Mean 1/λ, variance 1/λ², and it is memoryless.

MAT-101 · What is a cumulative distribution function?

F(x) = P(X ≤ x). It is non-decreasing, runs from 0 to 1, and for a continuous variable the PDF is f(x) = F′(x).

Statistics

MAT-102 · Mean, median and mode: which is robust to outliers?

The median. One extreme value can shift the mean a lot, but the median depends only on the middle position of the ordered data.

MAT-103 · What is standard deviation?

The square root of variance: σ = √(Σ(x − μ)²/N). It measures typical spread about the mean in the same units as the data.

MAT-104 · What is the difference between a population and a sample?

A population is the entire group of interest, described by parameters (μ, σ). A sample is a subset used to estimate them through statistics (x̄, s).

MAT-105 · What is the normal distribution?

A symmetric bell-shaped continuous distribution: f(x) = (1/(σ√(2π)))e^(−(x − μ)²/(2σ²)). Mean, median and mode are equal to μ.

MAT-106 · What is a z-score?

z = (x − μ)/σ, the number of standard deviations a value lies from the mean. It converts any normal variable to the standard normal N(0, 1).

MAT-107 · What is the correlation coefficient?

Pearson's r = Cov(X, Y)/(σ_X σ_Y), between −1 and +1. It measures the strength and direction of a linear relationship; 0 means no linear relationship.

MAT-108 · What is the difference between correlation and causation?

Correlation shows that two variables move together. Causation means one produces the other; it requires controlled experiments or causal reasoning, since confounders can create correlation.

MAT-109 · What is skewness?

A measure of asymmetry. Positive (right) skew has a long right tail with mean > median; negative skew has a long left tail with mean < median.

MAT-110 · What is kurtosis?

A measure of tail heaviness relative to the normal distribution, whose kurtosis is 3 (excess kurtosis 0). Higher kurtosis means more extreme outliers.

MAT-111 · What is simple linear regression?

Fitting y = a + bx by least squares, i.e. minimising the sum of squared vertical residuals, to predict y from x.

MAT-112 · What is a null hypothesis?

The default claim of no effect or no difference (H₀), tested against an alternative H₁. Data either give enough evidence to reject H₀ or fail to reject it.

MAT-113 · What is a p-value?

The probability, assuming H₀ is true, of getting a result at least as extreme as the one observed. If p is below the significance level α (often 0.05), H₀ is rejected.

MAT-114 · What is the interquartile range?

IQR = Q₃ − Q₁, the spread of the middle 50% of the data. It is robust to outliers; points beyond 1.5×IQR from the quartiles are commonly flagged as outliers.

Numerical Methods

MAT-115 · Why are numerical methods needed?

Many equations, integrals and ODEs have no closed-form solution or are too large to solve by hand. Numerical methods give approximate solutions to a controlled accuracy on a computer.

MAT-116 · What is the bisection method?

A root-finding method: if f(a)·f(b) < 0, a root lies in [a, b]. Evaluate the midpoint, keep the half where the sign changes, and repeat. It always converges for continuous f.

MAT-117 · What is the difference between truncation error and round-off error?

Truncation error comes from approximating an infinite process, such as cutting a Taylor series. Round-off error comes from the finite precision of computer arithmetic.

MAT-118 · What is the trapezoidal rule?

∫ₐᵇ f dx ≈ (h/2)[y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], with h = (b − a)/n. It approximates the curve by straight-line segments.

MAT-119 · What is Simpson's 1/3 rule?

∫ₐᵇ f dx ≈ (h/3)[y₀ + 4(y₁ + y₃ + …) + 2(y₂ + y₄ + …) + yₙ]. It fits parabolas through point triples and needs an even number of intervals n.

MAT-120 · What is Euler's method?

For y′ = f(x, y), step forward with y_(n+1) = yₙ + h·f(xₙ, yₙ). It is the simplest explicit ODE solver and is first-order accurate.

MAT-121 · What is Gaussian elimination?

A direct method for Ax = b that uses row operations to reduce A to upper triangular form, then solves by back-substitution. It costs about (2/3)n³ operations.

MAT-122 · What is interpolation?

Estimating values between known data points by constructing a function, usually a polynomial or spline, that passes exactly through them.

MAT-123 · What is the Newton–Raphson formula?

x_(n+1) = xₙ − f(xₙ)/f′(xₙ). It follows the tangent at xₙ to its x-intercept to get the next estimate of the root.

MAT-124 · What is the difference between absolute and relative error?

Absolute error = |true − approximate|. Relative error = absolute error/|true value|, often given as a percentage, and is independent of scale.

MAT-125 · What is the Gauss–Seidel method?

An iterative method for Ax = b that solves each equation for its diagonal unknown and immediately uses the newest values in the following equations. It usually converges faster than Jacobi.

MAT-126 · What is a finite difference?

A discrete approximation to a derivative from function values at nearby points, e.g. the forward difference f′(x) ≈ [f(x + h) − f(x)]/h.

Transforms (Laplace, Fourier, Z)

MAT-127 · What is the Laplace transform?

F(s) = ∫₀^∞ f(t)e^(−st) dt, with s a complex variable. It maps time-domain functions to the s-domain, where differentiation becomes multiplication by s.

MAT-128 · What are the Laplace transforms of 1, t and e^(at)?

L{1} = 1/s, L{t} = 1/s², L{tⁿ} = n!/sⁿ⁺¹, and L{e^(at)} = 1/(s − a) for s > a.

MAT-129 · What are the Laplace transforms of sin ωt and cos ωt?

L{sin ωt} = ω/(s² + ω²) and L{cos ωt} = s/(s² + ω²), for s > 0.

MAT-130 · What is a Fourier series?

A representation of a periodic function as a sum of sines and cosines: f(t) = a₀/2 + Σ[aₙ cos(nω₀t) + bₙ sin(nω₀t)], with ω₀ = 2π/T.

MAT-131 · What is the Fourier transform?

F(ω) = ∫₋∞^∞ f(t)e^(−iωt) dt. It decomposes a signal into a continuous spectrum of frequencies; the inverse is f(t) = (1/2π)∫F(ω)e^(iωt) dω.

MAT-132 · What is the Z-transform?

X(z) = Σ x[n]z⁻ⁿ for a discrete-time sequence x[n]. It is the discrete counterpart of the Laplace transform and is used for digital filters.

MAT-133 · What is the unit step function and its Laplace transform?

u(t − a) = 0 for t < a and 1 for t ≥ a. L{u(t)} = 1/s, and L{u(t − a)} = e^(−as)/s.

MAT-134 · What is the Dirac delta function and its transform?

δ(t) is zero for t ≠ 0, has unit area and satisfies ∫f(t)δ(t − a)dt = f(a). Its Laplace and Fourier transforms both equal 1.

MAT-135 · What is convolution?

(f*g)(t) = ∫ f(τ)g(t − τ)dτ. It gives the output of an LTI system to an input, and it becomes simple multiplication under Laplace, Fourier or Z transforms.

MAT-136 · How does symmetry affect a Fourier series?

An even function has only cosine terms (bₙ = 0). An odd function has only sine terms (a₀ = aₙ = 0). Half-wave symmetry removes even harmonics.

MAT-137 · What is the region of convergence (ROC)?

The set of s (Laplace) or z (Z-transform) values for which the transform sum or integral converges. It is needed to recover a unique time function and to judge stability and causality.

MAT-138 · How is an inverse Laplace transform usually found?

By splitting F(s) into partial fractions and matching each term with a standard transform pair from tables, together with the shifting theorems.

Discrete Math & Optimization

MAT-139 · What are the basic set operations?

Union A∪B (in either), intersection A∩B (in both), difference A − B (in A not B) and complement Aᶜ (not in A, relative to the universal set).

MAT-140 · What is a proposition and a truth table?

A proposition is a statement that is either true or false. A truth table lists the truth value of a compound statement for every combination of its components.

MAT-141 · What is a tautology?

A compound proposition that is true for every truth assignment, e.g. p ∨ ¬p. A contradiction is always false, e.g. p ∧ ¬p.

MAT-142 · What is the difference between permutations and combinations?

Permutations count ordered arrangements: ⁿPᵣ = n!/(n − r)!. Combinations count unordered selections: ⁿCᵣ = n!/(r!(n − r)!).

MAT-143 · What is a graph in discrete mathematics?

A set of vertices V connected by edges E. Graphs may be directed or undirected and weighted or unweighted, and they model networks, circuits and routes.

MAT-144 · What is a tree?

A connected graph with no cycles. A tree with n vertices has exactly n − 1 edges, and there is a unique path between any two vertices.

MAT-145 · What is an equivalence relation?

A relation that is reflexive (a R a), symmetric (a R b ⇒ b R a) and transitive (a R b, b R c ⇒ a R c). It partitions a set into equivalence classes.

MAT-146 · What are injective, surjective and bijective functions?

Injective (one-to-one): distinct inputs give distinct outputs. Surjective (onto): every element of the codomain is hit. Bijective: both, so an inverse exists.

MAT-147 · What is the pigeonhole principle?

If n + 1 or more objects are placed in n boxes, at least one box holds two or more objects. For example, among 13 people, two share a birth month.

MAT-148 · What is mathematical induction?

A proof method: show the statement holds for a base case n = n₀, then show that if it holds for n = k it holds for k + 1. It then holds for all n ≥ n₀.

MAT-149 · What is linear programming?

Optimizing (maximizing or minimizing) a linear objective function subject to linear equality and inequality constraints, used in production planning, scheduling and resource allocation.

MAT-150 · What is the difference between a local and a global optimum?

A local optimum is the best point in its neighbourhood. A global optimum is the best over the entire feasible region. For convex problems, every local minimum is global.

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