(SEM IV) THEORY EXAMINATION 2021-22 MATHS-IV

SECTION–A — Short Questions (But Conceptually Heavy)

Section–A consists of ten brief questions, each carrying two marks, but even though the questions are short, they cover a broad range of topics from partial differential equations, classification of PDEs, wave equations, probability distributions, regression, Bayes' theorem, F-test, and ANOVA. The paper begins with a straightforward first-order PDE p+q=1p + q = 1p+q=1, followed by a question on calculating the Particular Integral of a linear PDE involving operators DDD and D′D'D′, which requires recalling operator methods. One question asks you to classify the PDE 5uxx−9uxt+4utt=05u_{xx} - 9u_{xt} + 4u_{tt} = 05uxx​−9uxt​+4utt​=0, which tests your understanding of elliptic, parabolic, and hyperbolic equations.

The section also moves into applied mathematics by asking for the two-dimensional wave equation, and then shifts toward probability and statistics. You are asked to compute the moment generating function (MGF) of a negative exponential distribution, determine correlation using given regression coefficients, and compute the mean for a success/failure experiment involving rolling a die twice. The remaining questions deal with stating Bayes’ theorem, explaining when an F-test is used, and describing one-way ANOVA classification. Thus, Section–A tests the student’s ability to recall fundamental formulas, theorems, and definitions across PDEs and applied statistics while keeping the answers short and precise.
 

SECTION–B — Long, Descriptive, Analytical Questions (Any Three)

Section–B requires you to attempt any three of the five long questions, each worth ten marks. This section demands deeper explanation, step-wise derivations, and correct application of methods. One of the major questions asks you to solve a nonlinear first-order PDE using the Charpit method, a classical technique requiring computation of partial derivatives, characteristic equations, and integrating factors. Another question moves into heat conduction, asking you to solve the one-dimensional heat equation with zero boundary conditions and a sinusoidal initial condition u(x,0)=3sin⁡(πx/l)u(x,0) = 3\sin(\pi x/l)u(x,0)=3sin(πx/l). This requires applying separation of variables and understanding the physical meaning of decay in heat flow.

The paper also brings in real-world data analysis. One question provides paired x–y data and asks you to derive both regression lines — “y on x” and “x on y” — which involves computing means, deviations, regression coefficients, and formulating regression equations. Another statistical question deals with a large sample (2000 bulbs) whose lifetimes are normally distributed with a given mean and standard deviation. You must compute probabilities for bulbs lasting more than or less than certain lifetimes by converting values into Z-scores. The final question in this section involves applying the t-test to determine whether the sample mean differs significantly from an assumed mean, using the provided critical value of t. Thus, Section–B is a blend of PDE solving, classical methods, regression analysis, normal distribution applications, and hypothesis testing — requiring structured, logical, and detailed answers.
 

 SECTION–C — High-Level, Technical Application Questions (One From Each Set)

Section–C contains several groups of 10-mark questions where you must choose one from each set. These questions are more advanced and require strong problem-solving skills. The first set (Q3) deals with PDEs again: one question asks you to solve a second-order PDE involving mixed partial derivatives of zzz with respect to xxx and yyy, while the other asks you to solve a first-order PDE using the method of characteristics with the initial condition u(0,y)=sin⁡yu(0,y) = \sin yu(0,y)=siny. This tests your mastery over classical PDE solution techniques and ability to work with characteristic curves.

The next set (Q4) deals with separation of variables. One question gives a PDE ut−ux+2u=0u_t - u_x + 2u = 0ut​−ux​+2u=0 with a specific initial condition and asks you to solve it through decomposition into exponential terms. The alternate question involves solving Laplace’s equation with given boundary conditions, which requires constructing a Fourier series representation along one dimension while satisfying zero boundary conditions along others.

Further, Section–C includes questions on moments, skewness, and kurtosis (Q5a), which test your understanding of measures describing shape and symmetry of a distribution. The alternate (Q5b) asks you to fit a curve of the form y=c0+c1xy = c_0 + \frac{c_1}{\sqrt{x}}y=c0​+x​c1​​ using least squares, requiring you to reorganize the equation and minimize squared errors.

Next, Q6(a) tests your understanding of conditional probability and Bayes' theorem using a two-urn probability experiment involving white and blue balls. The alternate Q6(b) gives a real accident-frequency table over 50 days and asks you to fit a Poisson distribution, compute theoretical frequencies, and compare them with the observed pattern.

Finally, Q7(a) deals with a chi-square test for independence using real production data, requiring you to construct expected frequencies and compare them with observed values to see if demand changes across days of the week. The alternate Q7(b) gives data of defectives in 10 samples and asks you to construct a p-chart and determine whether the process is statistically in control — a typical quality-control question.

Thus, Section–C tests advanced understanding: solving PDEs, applying probability theory, using statistical inference, fitting distributions, constructing control charts, and performing hypothesis testing. All answers here must be methodical and mathematically precise.

Overall Summary (In Full Paragraph Form)

The Maths-IV examination paper is thoughtfully designed to test a wide range of mathematical competencies expected from a fourth-semester engineering student. Section–A checks basic understanding and quick recall across PDEs, transforms, probability distributions, and statistical tests. Section–B focuses on deeper analytical skills such as solving PDEs by classical methods, heat equation solutions, regression equation derivation, probability calculations using the normal distribution, and hypothesis tests like the t-test. Section–C is the most comprehensive, requiring strong mathematical maturity, as it covers advanced PDE solutions, Fourier-type boundary value problems, regression fitting, Bayes’ theorem, Poisson distribution modeling, chi-square testing, and quality-control charting. A student who is prepared in both theory and application will find this paper manageable and balanced, as it touches every major module of the syllabus with clarity and depth.