Polar Curves and Curvature: Polar coordinates, Polar curves, angle between the radius vector and the tangent, angle between two curves. Pedal equations. Curvature and radius of curvature – Cartesian, parametric, polar and pedal forms.

Series Expansion, Indeterminate Forms and Multivariable Calculus: Statement and problems on Taylor’s and Maclaurin’s series expansion for one variable. Indeterminate forms – L’Hospital’s rule. Partial differentiation, total derivative – differentiation of composite functions, Jacobian, Maxima and minima for the function of two variables.

Ordinary Differential Equations of First Order: Linear and Bernoulli’s differential equation. Exact and reducible to exact differential equations with integrating factors — 1/N (∂M/∂y − ∂N/∂x) and 1/M (∂N/∂x − ∂M/∂y). Orthogonal trajectories, Law of natural growth and decay.

Ordinary Differential Equations of Higher Order: Higher-order linear ordinary differential equations with constant coefficients, homogeneous and non-homogeneous equations (e^ax, sin(ax + b), cos(ax + b), xⁿ only). Method of variation of parameters, Cauchy’s and Legendre’s homogeneous differential equations. Applications: Solving governing differential equations of a mass–spring system.

Linear Algebra: Elementary row transformation of a matrix, Rank of a matrix. Consistency and Solution of system of linear equations – Gauss-elimination method and approximate solution by Gauss-Seidel method. Eigenvalues and Eigenvectors, Rayleigh’s power method to find the dominant Eigenvalue and Eigenvector. Applications: Traffic flow.