(SEM IV) THEORY EXAMINATION 2024-25 MATHEMATICS –IV

This document contains the complete B.Tech (Semester IV) Theory Examination 2024–25 question paper for MATHEMATICS–IV (BAS403). The paper spans four printed pages, carries 70 marks, and evaluates a student’s competency in Partial Differential Equations (PDEs), Fourier transforms, curve fitting, correlation & regression, probability distributions, numerical methods, hypothesis testing, and statistical decision-making.
 

Section A – Short Conceptual Questions (14 Marks)

Section A includes seven 2-mark questions designed to test fundamental mathematical understanding. Topics include:

Forming a partial differential equation by eliminating arbitrary functions

Solving a first-order PDE using the method of characteristics

Classifying a PDE operator

Finding mean & variance using given moments

Calculating regression coefficient

Finding probabilities from a joint density function

Writing the formula for Chi-square test

These questions test the foundational concepts of PDEs, statistics, and probability.
 

Section B – Intermediate Level Problems (21 Marks)

Students must attempt any three out of five questions. This section includes:

Solving a PDE using Lagrange’s equation

Solving the wave equation for a stretched string with fixed ends using given initial conditions

Computing absolute moments, β₁ and β₂ to comment on skewness and kurtosis

Finding the 99th percentile of a normal distribution using given mean, SD, and Z-values

Performing a Chi-square goodness-of-fit test for distribution of digits in a telephone directory

These questions assess analytical skills in PDEs, wave motion, descriptive statistics, and hypothesis testing.
 

Section C – Advanced PDEs, Fourier Transforms & Regression (35 Marks)

Section C contains five long questions, each with two subparts, covering:

1. Advanced PDE Solutions

Solving nonlinear PDEs using transformations

Solving operator-based PDEs such as (D−D′−1)(D−D′−2)z(D - D' -1)(D - D' - 2)z(D−D′−1)(D−D′−2)z

2. Heat Equation & Fourier Cosine Transform

Solving heat conduction problems for an insulated bar

Finding Fourier Cosine Transform of f(x)=11+x2f(x)=\frac{1}{1+x^2}f(x)=1+x21​

3. Curve Fitting & Regression

Fitting an exponential curve of the form y=abxy = ab^xy=abx

Finding lines of regression and estimating supply for a given price

4. Probability & Z-Distribution

Using binomial/Poisson approximation to calculate probabilities

Calculating number of students within score intervals using standard normal distribution

5. Hypothesis Testing & Control Charts

Performing a t-test to compare sample mean with assumed mean

Constructing an np-control chart to check if a manufacturing process is under control

These problems test mastery in higher mathematics, statistical inference, signal methods, and engineering-oriented PDE solutions.

Summary

This Mathematics–IV (BAS403) exam paper covers a comprehensive range of applied mathematics topics:

Formation & solution of PDEs

Method of characteristics & operator method

Wave & heat equations

Fourier transforms

Curve fitting, correlation & regression

Moments, skewness & kurtosis

Probability distributions & normal curve applications

Chi-square & t-tests

Control charts in quality control

It serves as a complete academic resource for engineering students studying advanced applied mathematics in their 4th semester.