(SEM II) THEORY EXAMINATION 2021-22 ENGINEERING MATHEMATICS-II

ENGINEERING MATHEMATICS-II | B.Tech (Semester II) | KAS203T

Theory Examination 2021–22 – Full Question Paper Overview

This document contains the complete and original B.Tech (Semester II) Theory Examination question paper for Engineering Mathematics–II (Subject Code: KAS203T) for the academic session 2021–22. The paper is structured to assess a student’s knowledge in advanced calculus, differential equations, multiple integrals, vector calculus, infinite series, complex analysis, contour integration, and conformal mappings—core components of second-semester engineering mathematics.

The question paper evaluates various learning levels:
Fundamental concepts
Analytical reasoning
Derivations
Higher-order problem-solving
Complex-variable techniques
Series expansions (Taylor, Laurent, Fourier)
Transformations in complex planes

The exam is divided into three major sections—A, B, and C—ensuring comprehensive evaluation.

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

Section A contains ten short questions, each testing precise conceptual understanding from different topics of Engineering Mathematics.

Key topics include:

Differential equation for a family of straight lines

Criterion for linear independence of solutions

Integral involving logarithmic functions

Volume generated by rotating an ellipse

Convergence of a trigonometric series

Fourier expansion of an even/odd piecewise function

Analyticity test for complex-valued functions

Geometric transformation under the mapping w=1zw = \frac{1}{z}w=z1​

Laurent series expansion about a non-zero point

Nature of singularities at finite and infinite points

These questions examine the student’s foundational clarity in calculus, complex analysis, and series.

SECTION B – Analytical & Problem-Solving Questions (3 × 10 = 30 Marks)

Students must attempt any three questions. This section contains advanced-level problems requiring multi-step reasoning, derivations, and mathematical modeling.

Topics covered include:

Solving simultaneous second-order differential equations

Application of the Gamma function identity

Testing convergence of series using ratio/root tests

Finding analytic functions from harmonic components using Cauchy–Riemann equations

Evaluating integrals using the method of contour integration

This section blends theoretical knowledge with problem-solving involving higher mathematics and complex-variable approaches.

SECTION C – Long & Higher-Order Thinking Questions (4 × 10 = 40 Marks)

Each subsection requires answering any one of the two long questions. These questions demand detailed understanding and step-by-step derivations.

Section C includes:

3. Differential Equation Methods

Variation of parameters

Differential equations with variable coefficients (e.g., Cauchy–Euler form)

4. Calculus of Multiple Integrals & Areas

Surface area generated by rotating a cardioid

Triple integrals with constraints in the positive octant

5. Infinite Series & Fourier Series

Convergence of general power series

Finding Fourier series for a piecewise function and deducing classical series results (e.g., 112+132+⋯=π28\frac{1}{1^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{8}121​+321​+⋯=8π2​)

6. Conformal Mapping & Bilinear Transformations

Mapping upper half-plane to upper half-plane

Finding bilinear transformations mapping three points

7. Complex Integration & Series Expansions

Evaluating complex integrals using Cauchy’s integral theorem and residue methods

Finding Taylor and Laurent series expansions in different regions (|z|<2, 2<|z|<3, |z|>3)

These questions evaluate the student's mastery over advanced mathematical tools used in engineering analysis.

Sample Questions (1–2 added as requested)

To represent the nature of the paper, here are sample questions included:

Sample Question 1 (from Section A):
“Find the differential equation which represents the family of straight lines passing through the origin.”
— Tests elimination of arbitrary constants.

Sample Question 2 (from Section C):
“Use the variation of parameters method to solve (D² – 1)y = 2(1 – e⁻²ˣ)⁻¹⁄².”
— Tests application of advanced differential equation-solving techniques.

These questions demonstrate the blend of conceptual and analytical depth throughout the paper.

Overall Coverage of the Question Paper

The topics in this question paper comprehensively cover:

Differential Equations

Homogeneous & non-homogeneous

Simultaneous differential equations

Variable coefficients

Variation of parameters

Advanced Calculus

Multiple integrals

Surface area and volume

Convergence tests

Fourier Series

Expansion on (−π,π)(-π, π)(−π,π)

Deduction of classical infinite series

Complex Analysis

Analyticity & Cauchy–Riemann conditions

Laurent and Taylor expansions

Contour integration

Conformal mapping

Bilinear transformations

Residue theorem

Special Functions & Gamma Functions

This structure ensures thorough evaluation of all critical chapters from the Engineering Mathematics-II syllabus.