(SEM III) THEORY EXAMINATION 2021-22 MATHEMATIVS-V

This is the B.Tech Semester III examination paper for MATHEMATICS-V (KAS304).
The paper carries 100 marks and evaluates the student’s knowledge of:

Fourier transforms

Z-transforms

Probability distributions

Hypothesis testing

Curve fitting

Numerical methods (Newton–Raphson, Regula Falsi, interpolation)

Statistical quality control (np-chart, p-chart, control limits)

Design of experiments (CRD, RBD, replication, randomization)

It is structured into three sections — A, B, and C, covering conceptual theory, analytical problems, and advanced statistical/numerical questions.

SECTION A – Short Questions (20 Marks)

This section contains 10 questions of 2 marks each, checking basic understanding:

Formula for Fourier sine transform of f′(x)f'(x)f′(x)

Z-transform of a constant kkk

Variance of Poisson distribution (mean = 4)

Definition of probability density function

Newton–Raphson formula

Relation between ∆ and ∇ in terms of shift operator EEE

Degrees of freedom for a 2×3 chi-square table

Meaning of null hypothesis

Objectives of experimental design

Difference between CRD (Completely Randomized Design) and RBD (Randomized Block Design)

These questions test basic recall and conceptual clarity.

SECTION B – Medium-Length Analytical Questions (30 Marks)

Students must attempt any three (each 10 marks).

Topics include:

(a) Fourier Transform Problem

Computing Fourier transform of a piecewise function 12a∣x∣≤a\frac{1}{2a}|x|\le a2a1​∣x∣≤a.

(b) Binomial Probability

10 bolts, 10% defective — find probability of:
(i) exactly one
(ii) none
(iii) at least one defective bolt.

(c) Lagrange Interpolation Polynomial

Construct the cubic polynomial passing through points:
(0,2)(0,2)(0,2), (1,3)(1,3)(1,3), (2,12)(2,12)(2,12), (5,147)(5,147)(5,147).

(d) ANOVA (F-test)

Using the deaths data of three hospitals (given on page 1), perform F-test to check if monthly deaths differ significantly.
(F-critical = 3.88 for 2,12 d.f. at 5% level.)

(e) Statistical Quality Control

Difference between p-chart and np-chart + construct np-chart for 10 samples of size 100 with defectives as given.

This section evaluates analytical ability, probability skills, interpolation knowledge, and statistical decision-making.

SECTION C – Long & Advanced Problems (50 Marks)

There are five main questions (Q3–Q7); students must attempt one part from each.

Q3 – Fourier Transform / Difference Equation

Fourier sine transform of e−ax/xe^{-ax}/xe−ax/x, a>0a>0a>0
OR

Solve the second-order difference equation yn+2+2yn+1+yn=ny_{n+2} + 2y_{n+1} + y_n = nyn+2​+2yn+1​+yn​=n with given initial conditions.

Q4 – Statistics (Normal Distribution / Discrete Distribution)

For resistors normally distributed with mean 100Ω, SD 2Ω — find % of resistors between 98Ω and 102Ω
OR

Find normalization constant kkk and mean for the probability function
f(x)=3k,2k,2k,kf(x)=3k,2k,2k,kf(x)=3k,2k,2k,k for x=0,1,2,3x=0,1,2,3x=0,1,2,3

Q5 – Numerical Methods

Solve x3−4x=9x^3 - 4x = 9x3−4x=9 using Regula-Falsi
OR

Fill missing table values using forward difference operator (table on page 2)

Q6 – Hypothesis Testing

Chi-square test to check if a die is biased (data given on page 2; χ² critical = 11.09)
OR

t-test for 9 sample values to check if mean differs significantly from assumed mean 47.5 (t-critical = 2.31)

Q7 – Experimental Design & Control Charts

Explain Replication, Randomization, Components of Variation
OR

Construct X-bar and R charts from given sample means & ranges of 10 samples (page 2)

This section tests statistical hypothesis testing, normal distribution analysis, numerical root-finding, and quality control chart construction.

OVERALL SUMMARY

The MATHEMATICS-V (KAS304) question paper provides a comprehensive assessment of:

Transform techniques (Fourier, Z-transform)

Probability theory & distributions (Poisson, binomial, PDFs)

Hypothesis testing (Chi-square, t-test, F-test)

Numerical analysis (Newton–Raphson, Regula-Falsi, interpolation)

Curve fitting & interpolation

Statistical quality control (control charts)

Design of experiments (CRD, RBD, replication, variation)

It combines theory, computation, statistics, numerical methods, and real-world data analysis—making it a complete and structured mathematical examination.