(SEM III) THEORY EXAMINATION 2023-24 MATHEMATICS-III

This examination assesses a student’s understanding of Partial Differential Equations (PDEs), Fourier Transforms, Numerical Methods, Regression Analysis, Moments, and Iterative Solving Techniques.
The paper is designed to test both conceptual clarity and problem-solving ability relevant to engineering applications.

The question paper is structured into three major sections, moving from basic definitions to advanced analytical problem-solving.

SECTION A – Short Answer / Conceptual Questions (14 Marks)

7 Questions × 2 Marks

This section examines fundamental mathematical concepts across PDEs, statistics, probability, and numerical analysis.

Key Topics Covered:

1. Solving PDE of the form pq = 3p + 4q

Tests understanding of first-order PDE formation and solution through standard methods.

2. Classification of PDE

Students must classify a second-order PDE (elliptic, parabolic, hyperbolic) using discriminant analysis.

3. Convolution Theorem of Fourier Transform

A theoretical question checking basic properties of Fourier transforms in engineering mathematics.

4. Binomial Distribution — Mean & Variance

Using mean = np and variance = npq to find missing binomial parameter q.

5. Value of Correlation from Regression Coefficients

Uses the relationship between regression coefficients and correlation coefficient:
 r=bxy⋅byx\, r = \sqrt{b_{xy} \cdot b_{yx} } r=bxy​⋅byx​​

6. Calculating Missing Term in a Difference Table

Concept of finite differences and interpolation.

7. Formula of Trapezoidal Rule

Tests knowledge of numerical integration techniques.

This section checks clarity of formulas, definitions, and small computations.

SECTION B – Descriptive / Analytical Questions (21 Marks)

Attempt any three (7 Marks each)
This section includes detailed problems from PDEs, heat equations, statistical estimation, and numerical methods.

Topics Covered:

1. Solving a PDE using Operator Notation

Uses differential operators DDD and D′D'D′ to solve non-homogeneous PDEs of the form (D2−2DD′+D′2)z=cos⁡(2y−3x)(D^2 - 2DD' + D'^2)z = \cos(2y - 3x)(D2−2DD′+D′2)z=cos(2y−3x).

2. One-Dimensional Heat Equation

Solving heat diffusion with boundary conditions u(0,t)=0, u(l,t)=0, u(x,0)=xu(0,t)=0,\ u(l,t)=0,\ u(x,0)=xu(0,t)=0, u(l,t)=0, u(x,0)=x using separation of variables.

3. Regression Equation Calculations

Compute the line of regression of y on x and x on y, including correlation coefficients, means, deviations, and regression constants.

4. Newton–Raphson Method

Finding a positive real value of 171/317^{1/3}171/3 correct up to 4 decimal places.

5. Fourth-Order Runge–Kutta Method

Solving the differential equation dy/dx=y+y2dy/dx = y + y^2dy/dx=y+y2 with y(0)=0y(0)=0y(0)=0 to find y(0.2)y(0.2)y(0.2).

This section evaluates stepwise reasoning, formula application, and computation skills.

SECTION C – Application-Based / Long Questions (35 Marks)

Students must answer one question from each sub-section.
These questions require deeper understanding and multi-step calculations.

C1 – Advanced PDE Solutions (7 Marks)

Option A – Solving Second-Order PDE

Solve

x2∂2z∂x2−y2∂2z∂y2=x2y2x^2\frac{\partial^2 z}{\partial x^2} - y^2\frac{\partial^2 z}{\partial y^2} = x^2y^2x2∂x2∂2z​−y2∂y2∂2z​=x2y2

using complementary and particular integrals.

Option B – Charpit’s Method

Solve the nonlinear PDE px+qy=pqpx + qy = pqpx+qy=pq.
This tests understanding of Charpit’s equations for first-order nonlinear PDEs.

C2 – Fourier Transform & Separation of Variables (7 Marks)

Option A – Fourier Sine Transform

Compute the sine transform of F(x)=e−axxF(x) = \frac{e^{-ax}}{x}F(x)=xe−ax​, a>0a>0a>0.

Option B – Solving PDE by Separation of Variables

Solve

ut−ux+2u=0,u(x,0)=10e−x−6e−4xu_t - u_x + 2u = 0,\quad u(x,0)=10e^{-x}-6e^{-4x}ut​−ux​+2u=0,u(x,0)=10e−x−6e−4x 

C3 – Statistical Curve Fitting & Moments (7 Marks)

Option A – Least Squares Method

Fit y=a+bxy = a + bxy=a+bx to given data using normal equations.

Option B – Skewness & Kurtosis

Using the first four central moments (1, 4, 10, 45) about 4, compute measures of asymmetry and peakedness.

C4 – Numerical Approximation Methods (7 Marks)

Option A – Regula Falsi (False Position Method)

Find root of 5x−3x3=05x - 3x^3 = 05x−3x3=0 correct to 4 decimals.

Option B – Newton’s Divided Difference Interpolation

Using the given x and f(x), calculate f(6)f(6)f(6).

C5 – System Solving / Numerical Integration (7 Marks)

Option A – Gauss–Seidel Method

Solve the linear system:

{2x+10y+z=5110x+y+2z=44x+2y+10z=61\begin{cases} 2x + 10y + z = 51 \\ 10x + y + 2z = 44 \\ x + 2y + 10z = 61 \end{cases}⎩⎨⎧​2x+10y+z=5110x+y+2z=44x+2y+10z=61​

Option B – Simpson’s One-Third Rule

Evaluate

∫0611+x2dx\int_0^6 \frac{1}{1+x^2} dx∫06​1+x21​dx

using numerical integration.

Purpose of This Examination

This Mathematics–III exam measures students’ ability to:

Solve complex PDEs used in heat flow, wave motion, and engineering physics
Apply Fourier techniques for signal processing and transforms
Use numerical methods for real-world approximations
Compute regression, correlation, skewness, and kurtosis
Apply iterative root-finding and interpolation techniques
Demonstrate analytical reasoning and problem-solving accuracy

It builds mathematical competence essential for engineering analysis, modeling, simulation, and computation.