(SEM IV) THEORY EXAMINATION 2023-24 MATHEMATICS-V

The uploaded document is the B.Tech (Semester IV) Theory Examination Paper for the academic year 2023–24 for the subject BAS404 – Mathematics-V. It is an official university examination paper that comprehensively evaluates a student’s understanding of advanced engineering mathematics, including transforms, numerical methods, probability distributions, hypothesis testing, and statistical quality control. The exam carries 70 marks, must be completed in 3 hours, and contains questions spread across Section A, Section B, and Section C, as shown on Page 1 and Page 2 of the file.
 

The paper begins with Section A, which contains seven compulsory short-answer questions, each worth 2 marks, designed to test foundational knowledge. Students must define a Fourier transform, find the Laplace transform of t·sin t, and state the condition under which a binomial distribution becomes a Poisson distribution, which introduces the convergence concept in probability. Another question requires writing the divided-difference formula for a function defined at three points x0,x1,x_0, x_1,x0​,x1​, and x2x_2x2​, reflecting the basics of interpolation theory. Students are also asked to show the relation Δ=∇E\Delta = \nabla EΔ=∇E, demonstrating their understanding of shift operators in finite difference calculus. The definition of a chi-square test is required, testing their knowledge of hypothesis testing, while the final question asks for the objective of experimental design, reflecting the role of statistical planning in data analysis. Section A ensures that students possess essential mathematical fundamentals before progressing to more complex topics.
 

Section B contains five descriptive questions, out of which any three must be attempted, each carrying 7 marks. These questions require deeper explanation, step-by-step reasoning, and sometimes numerical computations. One question asks the student to explain the Z-transform and the modulation property of the Fourier transform, demonstrating how signals behave in transform domains. Another question presents a probability distribution table and asks students to compute the value of kkk, the mean, and the variance, evaluating their ability to normalize and analyze discrete probability distributions. A detailed explanation of the Regula-Falsi method—along with its applications and limitations—is also required, demonstrating the student’s understanding of root-finding techniques. A statistical question asks whether the nature of an area (rural or urban) is related to voting preference, supported by a contingency table on Page 1, requiring students to perform a chi-square test of independence. The final question in this section presents 15 sample plots across three varieties, asking if there is significant variation using F-test (ANOVA), testing the student’s understanding of variance analysis and group comparison.
 

Section C contains long, analytical questions divided into subsections (Q3 to Q7), and students must attempt one question from each subsection, with each question carrying 7 marks. The first subsection asks students either to find the Fourier cosine transform of the function f(x)=e−xxf(x) = \frac{e^{-x}}{x}f(x)=xe−x​ or to solve a difference equation using the Z-transform, requiring high-level command over transforms and sequence analysis.

The next subsection contains a classical probability question involving horse-kick deaths in 200 army corps over 10 years. Students must determine the corresponding Poisson distribution function and compute theoretical frequencies, showing how real-life data can be fitted to discrete probability models. Alternatively, students may answer a question asking how many times at least seven heads are expected when 10 coins are tossed 100 times, testing their ability to compute binomial probabilities.

The next subsection focuses on numerical methods. One option asks students to explain Lagrange interpolation and the divided difference formula, evaluating their understanding of interpolation theory. The alternative requires proving that the Newton–Raphson method has second-order convergence, demonstrating deeper insight into numerical error analysis.

Another subsection asks whether there is an association between area and pollution index using a chi-square test, based on the contingency table given on Page 2. The alternative asks students to explain the t-test for one and two samples, covering assumptions, test statistics, and significance interpretation.

The final subsection deals with statistical quality control and experimental design. One question asks students to explain the terms RBD (Randomized Block Design), LSD (Latin Square Design), and CRBD, illustrating their use in experimental planning. The alternative question asks them to discuss the difference between np-chart and p-chart, and then construct an np-chart using the 10 sample values provided on Page 2, showing their ability to apply control-chart methodology to real defect data.

Overall, the Mathematics-V exam paper provides a complete and rigorous evaluation of theoretical, computational, and applied mathematical skills. It tests transform techniques, interpolation, numerical root-finding, probability distributions, statistical hypothesis testing, ANOVA, regression concepts, and quality-control tools—making it an essential part of engineering mathematics education. The combination of short, medium, and long analytical questions ensures that the student demonstrates clarity of concepts, strong reasoning, numerical accuracy, and problem-solving maturity.