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 ∫sin4θ 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.
