Exam
## Exam

**XE-A (Compulsory for all XE candidates) Engineering Mathematics**

**Section 1: Linear Algebra**

Algebra of matrices; Inverse and rank of a matrix; System of linear equations; Symmetric,

skew-symmetric and orthogonal matrices; Determinants; Eigenvalues and eigenvectors;

Diagonalisation of matrices; Cayley-Hamilton Theorem.

**Section 2: Calculus**

Functions of single variable: Limit, continuity and differentiability; Mean value theorems;

Indeterminate forms and L'Hospital's rule; Maxima and minima; Taylor's theorem;

Fundamental theorem and mean value-theorems of integral calculus; Evaluation of

definite and improper integrals; Applications of definite integrals to evaluate areas and

volumes.

Functions of two variables: Limit, continuity and partial derivatives; Directional derivative;

Total derivative; Tangent plane and normal line; Maxima, minima and saddle points;

Method of Lagrange multipliers; Double and triple integrals, and their applications.

Sequence and series: Convergence of sequence and series; Tests for convergence;

Power series; Taylor's series; Fourier Series; Half range sine and cosine series.

**Section 3: Vector Calculus**

Gradient, divergence and curl; Line and surface integrals; Green's theorem, Stokes

theorem and Gauss divergence theorem (without proofs).

**Section 3: Complex variables**

Analytic functions; Cauchy-Riemann equations; Line integral, Cauchy's integral theorem

and integral formula (without proof); Taylor's series and Laurent series; Residue theorem

(without proof) and its applications.

**Section 4: Ordinary Differential Equations**

First order equations (linear and nonlinear); Higher order linear differential equations with

constant coefficients; Second order linear differential equations with variable

coefficients; Method of variation of parameters; Cauchy-Euler equation; Power series

solutions; Legendre polynomials, Bessel functions of the first kind and their properties.

**Section 5: Partial Differential Equations**

Classification of second order linear partial differential equations; Method of separation

of variables; Laplace equation; Solutions of one dimensional heat and wave equations.

**Section 6: Probability and Statistics**

Axioms of probability; Conditional probability; Bayes' Theorem; Discrete and continuous

random variables: Binomial, Poisson and normal distributions; Correlation and linear

regression.

**Section 7: Numerical Methods**

Solution of systems of linear equations using LU decomposition, Gauss elimination and

Gauss-Seidel methods; Lagrange and Newton's interpolations, Solution of polynomial and

transcendental equations by Newton-Raphson method; Numerical integration by

trapezoidal rule, Simpson's rule and Gaussian quadrature rule; Numerical solutions of first

order differential equations by Euler's method and 4th order Runge-Kutta method.

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