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

The uploaded document is the B.Tech Semester IV Theory Examination Paper (2023–24) for the subject BAS402 – Mathematics–III. This official university examination paper evaluates a student’s understanding of advanced mathematical methods used in engineering, including partial differential equations, Fourier analysis, probability theory, numerical methods, and iterative computational techniques. The paper carries 70 marks, must be completed in 3 hours, and is structured into clearly defined sections. All questions are presented in English, and the content across Page 1 and Page 2 is cleanly formatted with equations, numerical values, and problem statements that require analytical, step-by-step solutions.
 

The paper begins with Section A, which contains ten compulsory short-answer questions, each worth 2 marks, meant to test fundamental conceptual knowledge. These questions include writing partial differential equations from given functions, defining complementary functions and particular integrals, solving basic probability questions such as finding the mean and variance of a distribution, and writing formulas for Fourier sine and cosine transforms. Students are also asked to classify partial differential equations into elliptic, parabolic, or hyperbolic forms, and to state methods such as the Gauss–Seidel iteration or central difference schemes. Section A ensures that students have essential foundational clarity in PDEs, numerical methods, and probability before they move to more comprehensive questions.

Section B contains five descriptive questions, out of which students must attempt any three, each carrying 7 marks. These questions require detailed explanations, derivations, stepwise problem solving, and conceptual clarity. One question involves solving a first-order linear partial differential equation using Lagrange’s method, which demands proper formation of auxiliary equations and integration of characteristic curves. Another question asks students to compute the Fourier sine transform of a function and derive its inverse, emphasizing the role of Fourier analysis in boundary-value problems. The section also includes numerical techniques such as applying Simpson’s 1/3 rule for definite integrals or solving systems of linear equations using the Gauss–Seidel iterative method, requiring multiple iterations and tabular representation. Students may also be asked to compute correlation coefficients, perform interpolation using Newton’s forward or backward methods, or solve probability questions involving normal distribution. These questions test practical application and computational proficiency.

Section C contains long, analytical questions divided into several subsections (Q3 to Q7), each requiring students to attempt one question, with each question carrying 7 marks. These questions focus on advanced derivations, multi-step solutions, and interpretation of methods. One subsection asks students to solve the one-dimensional wave equation using separation of variables, requiring formulation of boundary and initial conditions, characteristic solution and final representation in terms of sine series. Another question explores solving Laplace or Poisson equations, testing their understanding of boundary-value problems in engineering domains.

In the numerical-methods part, students may be asked to apply Regula–Falsi (False Position) method to find the root of a nonlinear equation, showing iterations until convergence. Another option includes using the Runge–Kutta 4th-order method to solve an ordinary differential equation numerically, requiring tabulated values and approximation for the next step. Students may also solve a system of linear simultaneous equations using matrix inversion or LU decomposition, demonstrating proficiency in linear algebraic techniques.

Probability and statistics questions in Section C may include evaluating chi-square values, testing hypotheses, or interpreting normal distribution scenarios, as shown in the numerical table printed on Page 2 of the exam. Students are required to demonstrate understanding of statistical testing, variance computation, and probability-density interpretations.

Overall, the Mathematics–III examination paper is a comprehensive and rigorous assessment tool that combines theory, application, and computation. It tests fundamental definitions, PDE formation, transforms, differential equations, vector calculus, Fourier analysis, interpolation, numerical integration, iterative algorithms, and probability-distribution problems. The balanced mix of conceptual, numerical, derivation-based, and application-oriented questions ensures that students demonstrate both mathematical maturity and problem-solving accuracy. The exam structure—divided into short, medium, and long descriptive questions—creates a complete and thorough evaluation of the student’s understanding of Mathematics–III.