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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.