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.
