(SEM II) THEORY EXAMINATION 2018-19 MATHEMATICS-II

This document contains the complete and authentic B.Tech (Semester II) Mathematics-II Theory Examination Question Paper, as prescribed by the university for the academic session. The paper follows the official structure defined in the syllabus and evaluates a student's conceptual clarity, problem-solving ability, analytical skills, and understanding of advanced mathematical tools used in engineering.

The question paper covers a comprehensive range of topics from Differential Equations, Multiple Integrals, Gamma & Beta Functions, Fourier Series, Complex Analysis, Conformal Mappings, Cauchy–Riemann Conditions, and Residue Calculus. It consists of three major sections, designed to test increasing levels of difficulty—starting from basic definitions to high-level analytical reasoning.

SECTION A – Short Answer Questions (10 × 2 = 20 Marks)

This section contains 10 short conceptual questions designed to test fundamental understanding.
Topics include:

Finding particular integrals of differential equations

Solving simultaneous differential equations

Volumes of solids of revolution

Gamma function evaluations

Fourier constants

Convergence of sequences

Analyticity of complex functions

Conformal mapping definition

Complex integration along curves

Finding residues at singularities

These questions evaluate clarity of concepts and quick application of standard formulas.

SECTION B – Descriptive Problems (Any 3 × 10 = 30 Marks)

This section contains advanced mathematical questions that require derivations, calculations, and method-based solutions.

Topics include:

Frobenius method for solving differential equations

Dirichlet integrals and volume of sphere octants

Half-range sine series expansion

Harmonic functions & analytic function reconstruction using Cauchy–Riemann equations

Laurent series expansions in different regions

These questions test deeper analytical skills, mathematical reasoning, and strong command of classical calculus and complex analysis.

SECTION C – Long Answer / Analytical Questions (4 × 10 = 40 Marks)

Each subsection has two choices; students must solve one.

Major topics include:

Differential Equations

Variation of parameters

Normal form method

Integral Calculus, Beta & Gamma Functions

Proving Gamma & Beta function relations

Solving integrals using Beta–Gamma identities

Fourier Series & Convergence Tests

Full Fourier series expansion

D’Alembert’s test for convergence

Complex Analysis

Cauchy–Riemann conditions at singular points

Möbius (Bilinear) transformations

Cauchy Integral & Residue Calculus

Cauchy integral formula evaluation

Complex integrals using residue theorem

This section requires detailed solutions, step-by-step reasoning, clear justification, and high-level mathematical understanding.

Sample Questions from the Paper

Here are two representative questions from the uploaded paper:

Example 1 (Section A)

“Find residue of f(z)=ez−1z3f(z)=\frac{e^z -1}{z^3}f(z)=z3ez−1​ at z=0z = 0z=0.”
— Tests knowledge of singularities and residue evaluation.

Example 2 (Section C)

“Use variation of parameters to solve y′′+y=tan⁡xy'' + y = \tan xy′′+y=tanx.”
— Checks advanced differential equation solving skills.

Overall Coverage and Purpose of This Question Paper

The uploaded Mathematics-II exam paper 

MATHEMATICS-II-KAS-203 (1)

 comprehensively evaluates the following:

Differential Equations

Homogeneous and non-homogeneous ODEs, Frobenius method, normal forms.

Advanced Integration & Special Functions

Multiple integrals, Beta & Gamma functions, Dirichlet integrals.

Fourier Series & Convergence

Full and half-range expansions, convergence testing.

Complex Variables

Analytic functions, Cauchy–Riemann conditions, conformal mapping.

Laurent Series & Residue Calculus

Singularities, contour integrals, Möbius transformations.