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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.

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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.

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Section 3: Vector Calculus

Gradient, divergence and curl; Line and surface integrals; Green's theorem, Stokes
theorem and Gauss divergence theorem (without proofs).

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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.

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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.

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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.

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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.

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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.