(SEM IV) THEORY EXAMINATION 2017-18 MATHEMATICS - III

The B.Tech Mathematics-III (RAS401) question paper from the 2017–18 IV Semester Theory Examination is a well-structured, 70-mark examination designed to test engineering students on advanced mathematical concepts essential for higher technical studies. The paper spans complex analysis, numerical methods, transforms, probability & statistics, and differential equations, providing comprehensive evaluation through conceptual, analytical, and application-based questions. 

SECTION A – Fundamental Concepts (Short Answer Questions)

This section consists of 7 brief questions (14 marks) requiring precise definitions, formulae, and conceptual clarity. The topics include:

Types of singularities in complex analysis

Cauchy–Riemann equations in polar form

Normal distribution application problem on army shoes’ life expectancy

Moment generating function of Binomial distribution

A proof involving expectation notation

Newton–Cotes Quadrature formula for numerical integration

Z-transform of a step function u(−k)

This section assesses basic theoretical understanding across different areas of engineering mathematics. 

SECTION B – Intermediate Analytical Problems (3 × 7 = 21 Marks)

Students must attempt any three questions, each requiring detailed calculations or derivations.

The section includes problems such as:

Determination of an analytic function from given u and v relations

Mean and variance of a Poisson distribution

Numerical integration of ∫06ex1+xdx\int_0^6 \frac{e^x}{1 + x} dx∫06​1+xex​dxusing:

Trapezoidal rule

Simpson’s 1/3 rule

Simpson’s 3/8 rule

Angular velocity and acceleration calculation from the data table (θ vs. time) on page 1

Fourier cosine transform and hence the Fourier sine transform of 11+x2\frac{1}{1 + x^2}1+x21​

The table given for rotational motion on page 1 visually presents time intervals and angle measurements, enabling numerical differentiation. 

SECTION C – Advanced Theory, Derivations & Numerical Methods (5 Questions × 7 Marks)

Each question has “Attempt any one part” choice, and covers complex problem-solving, verification of theorems, numerical root solving, matrix decomposition, Z-transforms, and differential equations.

Key topics include:

Q3 – Complex Analysis Applications

Verification of Cauchy’s Theorem by integrating eize^{iz}eiz around a triangular contour with vertices (1+i, –1+i, –1–i)

Evaluation of a definite integral ∫0∞sin⁡mxxdx\int_0^{\infty} \frac{\sin mx}{x} dx∫0∞​xsinmx​dx (a standard Fourier-type integral)
 

Q4 – Statistics & Curve Fitting

Calculation of skewness and kurtosis from the height distribution of 100 students (table visible on page 2)

Least squares curve fitting for the function y=c0x+c1xy = \frac{c_0}{x} + c_1\sqrt{x}y=xc0​​+c1​x​ using 

Q5 – Roots of Equations & Interpolation

Solving xex=cos⁡xx e^x = \cos xxex=cosx using the Regula–Falsi method (accurate to 4 decimal places)

Construction of Newton’s divided difference polynomial from given data points
(x = –3, –1, 0, 3, 5) and corresponding f(x) values
 

Q6 – Differential Equations & Linear Algebra

Solving the IVP u′=–2tu2,  u(0)=1u' = –2tu^2, \; u(0) = 1u′=–2tu2,u(0)=1 using RK-4 method (step size h = 0.2, interval [0, 0.4])

Solving a system of linear equations using LU decomposition with lii=1l_{ii}=1lii​=1
 

Q7 – Z-Transform & Heat Equation

Solving a difference equation using Z-transform method with given initial conditions y0=0,y1=1y_0 = 0, y_1 = 1y0​=0,y1​=1

Solving the heat equation in a semi-infinite rod 0≤x<∞0 ≤ x < ∞0≤x<∞ using PDE methods and boundary conditions provided (including Neumann condition at x=0)

OVERALL ESSENCE OF THE PAPER

The Mathematics-III question paper comprehensively evaluates students on:
Complex analysis & analytic functions
Probability distributions & statistical measures
Fourier transforms
Numerical integration & differentiation
Curve fitting & interpolation
Ordinary differential equations (ODEs)
Partial differential equations (PDEs) in heat transfer
Numerical root finding
Z-transform techniques
Matrix decomposition for solving linear systems

This paper is ideal for exam preparation, mathematical revision, and understanding the structure of B.Tech-level mathematical assessments.