(SEM III) THEORY EXAMINATION 2017-18 Mathematics -III

This document contains the complete B.Tech (Semester III) Theory Examination 2017–18 question paper for Mathematics–III (RAS-301). The paper consists of two printed pages, carries 70 marks, and evaluates a student’s understanding of complex analysis, numerical methods, Fourier transforms, Z-transforms, probability distributions, interpolation, correlation, regression, and numerical integration.

Section A – Short Conceptual Questions (14 Marks)

Section A contains seven brief questions, each designed to test foundational mathematical knowledge. Topics include:

Definition and example of an analytic function

Binomial distribution with its mean and variance

Normal equation for curve fitting of the form ya=bx+xay_a = bx + x_aya​=bx+xa​

Comparison between Regula Falsi and Newton–Raphson methods

Relationship between nth divided difference and nth forward difference

Meaning of an initial value problem

Finding the inverse Z-transform of a given expression

These questions ensure that students understand the fundamental concepts needed for higher mathematical analysis.
 

Section B – Descriptive & Problem-Solving Questions (21 Marks)

Students must attempt any three questions involving proofs, calculations, and analytical reasoning:

Example of a function satisfying Cauchy–Riemann equations but not analytic at the origin

Calculation of skewness and kurtosis using moments for a given distribution

LU decomposition of matrix A and solving a 3×3 system of linear equations

Expressing a piecewise function using Fourier Integral and evaluating a definite integral

Solving an initial value problem using Runge–Kutta 4th order method up to x=0.6x = 0.6x=0.6

This section assesses logical reasoning, numerical accuracy, and deeper conceptual understanding.
 

Section C – Advanced Long Questions (35 Marks)

This section contains five groups, each requiring students to attempt one part only. Topics include:

1. Complex Analysis

Evaluation of ∫sin⁡4θ dθ\int \sin^4 \theta \, d\theta∫sin4θdθ

Cauchy Integral Formula with evaluation of ∫Cdz(z2+5)2\int_C \frac{dz}{(z^2+5)^2}∫C​(z2+5)2dz​

2. Fourier & Z-Transforms

Fourier cosine transform of 11+x2\frac{1}{1+x^2}1+x21​ and hence sine transform of x1+x2\frac{x}{1+x^2}1+x2x​

Inverse Z-transform using partial fractions and long division

3. Probability & Statistics

Finding mean, standard deviation, and coefficient of correlation for partially given data

Mean and variance of the normal distribution

4. Numerical Methods

Finding a real root of x3−2x+5=0x^3 - 2x + 5 = 0x3−2x+5=0 using the False Position method

Lagrange interpolation formula and constructing the interpolating polynomial

5. Numerical Integration & Maxima

Applying Simpson’s 3/8 rule and Simpson’s 1/3 rule to evaluate definite integrals

Finding the value of x for which y is maximum using tabulated data

These questions test mastery over calculus, numerical analysis, integration techniques, and statistical interpretation.
 

Summary

The Mathematics-III (RAS-301) exam paper comprehensively evaluates:

Complex variables & analytic functions

Fourier integrals & transforms

Z-transform and inverse Z-transform

Probability distributions & statistical measures

Curve fitting, correlation & regression

Numerical methods: RK-4, Newton–Raphson, Regula-Falsi, interpolation

Numerical integration (Simpson’s methods)

It is an essential academic resource for engineering students studying advanced applied mathematics in the third semester.