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

The uploaded document is the B.Tech Semester IV Theory Examination Paper (2023–24) for the subject BAS403 – Mathematics–IV. This official university examination paper is designed to assess a student’s understanding of higher engineering mathematics, including partial differential equations, Fourier transforms, regression analysis, probability theory, curve fitting, statistical quality control, hypothesis testing, and sampling distribution problems. The paper carries 70 marks, is structured for 3 hours, and is fully bilingual in English and Hindi, as clearly shown across all pages (Page 1 to Page 4) of the uploaded file.

The examination begins with Section A, which contains seven compulsory short-answer questions, each worth 2 marks, aimed at testing basic conceptual clarity. The questions include defining a partial differential equation from a given function, writing the complementary function for a PDE, and defining skewness and kurtosis, which are important measures of shape in statistics. Students are also asked to derive a regression line and demonstrate that the covariance of an independent variable with its error term is zero, ensuring theoretical correctness for least squares estimation. Another question tests understanding of how the Poisson distribution can be derived as a limiting case of the binomial distribution. A final question requires students to write the analytical solution of a wave equation with specified boundary conditions. Section A ensures that students grasp the essential definitions, facts, and general formulas before moving into more detailed problem solving.

Section B contains five descriptive questions, and students must attempt any three, each worth 7 marks. This section requires full derivations, reasoning, and sometimes diagrammatic or numerical demonstration. One question asks students to apply Charpit’s method to solve a first-order nonlinear partial differential equation, requiring the construction and integration of characteristic curves. Another question asks students to find the Fourier sine transform of a function and derive the corresponding inverse transform, highlighting the importance of Fourier analysis in solving PDEs. There is a question on solving a wave equation for a taut string fastened at both ends, requiring the method of separation of variables and the imposition of boundary conditions. Another question asks for the derivation of regression lines and a proof that the regression coefficients satisfy the relation byxbxy=r2b_{yx} b_{xy} = r^2byx​bxy​=r2, linking regression to the correlation coefficient. The final question in this section requires students to demonstrate how the Poisson distribution emerges as a limiting case of the binomial distribution when the number of trials becomes very large and the probability of success very small, illustrating a fundamental result in probability theory.

Section C contains analytical questions grouped into subsections (Q3 to Q7). Students must answer one question from each subsection, with each question carrying 7 marks. The first subsection asks students to explain control charts, particularly p-charts and np-charts, used in statistical quality control. The alternate question asks for the equation of the binomial test, explaining how it is used to test hypotheses about proportions. Another subsection asks students to fit a curve of the form y=axby = a x^by=axb using the method of least squares, which requires converting the equation into a logarithmic linear form and solving for parameters. Another question asks students to calculate the coefficient of skewness and explain its role in indicating whether a distribution is positively skewed, negatively skewed, or symmetric.

In the probability and hypothesis testing section, students are asked to work with a normal distribution problem using mean and standard deviation, using Z-table values printed on the question paper to compute probabilities of certain ranges. Alternatively, they may be asked to conduct a chi-square test for independence or association between two categorical variables, determining whether the observed frequencies deviate significantly from expected frequencies. The following subsection tests whether sample means from two populations differ significantly by using the large-sample Z-test, involving computation of standard error and the test statistic. The alternate asks whether there is a significant difference between two population variances using the F-test, which is widely used in variance comparison problems.

Finally, the last subsection requires students to find the mean of a binomial distribution, demonstrating the derivation of E(X)=npE(X) = npE(X)=np, or to prove the relation for the moment-generating function of a binomial distribution. Alternatively, they may compute probabilities using the Poisson distribution as a discrete probability model.

Overall, the exam paper for Mathematics–IV provides a complete and rigorous evaluation of theoretical, computational, and application-based mathematical skills. It systematically examines PDEs, transformations, probability, regression, curve fitting, hypothesis testing, and sampling distributions — all essential tools in higher engineering studies. The bilingual format, precise structure, mix of conceptual and analytical questions, and coverage of both classical and modern mathematical methods make this paper a comprehensive assessment of the student’s understanding of Mathematics–IV.