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

SECTION–A (Short Questions but Deep Concepts – 20 Marks)

Section–A of this paper contains ten questions, each worth two marks. Although these questions are categorized as “brief,” they actually cover the foundational theories of Laplace transforms, Z-transforms, Fourier transforms, relations, group theory, irrational numbers, inverse functions, Boolean algebra, and lattice theory. Even though answers are short, the examiner is checking whether you understand fundamental definitions, properties, and proofs.

For instance, the very first question asks you to evaluate the Laplace transform of e−tcos⁡te^{-t} \cos te−tcost, which tests whether you remember standard Laplace formulas and how shifting works. Another question asks you to state the convolution theorem, which is crucial for solving Laplace-based differential equations. You are also asked about the Z-transform of a cosine sequence, the change-of-scale property in Fourier transform, and an example of an equivalence relation — all of which test your conceptual clarity.

Then Section–A transitions into abstract algebra, where you must show that every cyclic group is abelian or prove that 7\sqrt{7}7​ is irrational using contradiction. These require short but logically sound reasoning. Questions such as finding the inverse of a linear function and explaining idempotent laws check your ability to apply algebraic rules. Finally, describing a complemented lattice tests your understanding of discrete mathematical structures. So, while this section is short in length, it is heavy in fundamental ideas and tests whether you can recall and express core mathematical properties clearly.

SECTION–B (Three Long Answers – 30 Marks)

Section–B asks you to attempt any three out of five 10-mark questions. These questions require thorough explanations, derivations, and step-by-step reasoning. The first question deals with the Laplace transform of a periodic triangular wave, which requires understanding periodic extensions and how to apply Laplace formulas for piecewise functions. Another question uses Z-transform to solve a difference equation, which means you must convert recurrence relations into algebraic equations in the Z-domain, solve them, and then apply inverse Z-transform.

You are also asked to state and prove Lagrange’s Theorem from group theory, one of the most important theorems in algebra relating group order and subgroup order. Another question asks you to solve a recurrence relation an=2an−1−3a_n = 2a_{n-1} - 3an​=2an−1​−3 with an initial condition, requiring you to apply linear recurrence techniques or homogeneous solution methods. The final option in this section involves Boolean algebra, where you must simplify a Boolean expression using K-map and then explain AND, OR, and NOT gates and draw their circuits. This section expects students to show mathematical maturity, logical structuring of answers, and competency in both continuous and discrete mathematics.

SECTION–C (Higher-Order Application Questions – 10 Marks Each)

Section–C contains multiple groups of advanced, 10-mark questions. From each group, you must attempt one part. These questions require deeper mathematical thinking, strong problem-solving skills, and the ability to use transforms, PDE solving techniques, logic, induction, and set theory.

In the first group (Q3), one option asks you to solve a pair of simultaneous differential equations using Laplace transform. This tests your ability to convert a system of differential equations into algebraic equations using Laplace rules and then return to time-domain solutions. The alternate option asks for the inverse Laplace transform of a logarithmic function, which requires careful use of transform tables and sometimes differentiation in the s-domain.
 

The next group (Q4) moves into partial differential equations. One question gives the heat equation ∂u∂t=∂2u∂x2\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}∂t∂u​=∂x2∂2u​ with specific boundary and initial conditions. This requires solving using Fourier series or separation of variables. The alternate option uses residue theory of complex variables to find an inverse Z-transform, which is a higher-level concept needing contour integration ideas.
 

In the group (Q5), Section–C enters logic and reasoning. One question asks you to test the validity of a multi-step argument involving statements about weather, swimming, canoeing, and going home early. This requires forming propositions, identifying implications, and verifying whether the final conclusion logically follows. The alternate option asks you to prove a logical equivalence using algebra of propositions.

Next (Q6), the paper tests induction and set counting. One option requires proving a long summation identity using mathematical induction, while the other involves inclusion–exclusion principle to count programmers who are not proficient in any of three languages (Java, C#, Python). Both questions require very clear step-based reasoning.

Finally, the last group (Q7) focuses on posets (partially ordered sets). One question asks you to prove that a poset can have at most one greatest and one least element. The other option asks you to define least and greatest elements, describe the power set of S={a,b,c}S = \{a, b, c\}S={a,b,c}, draw the Hasse diagram for (P(S),⊆)(P(S), \subseteq)(P(S),⊆), and identify all special elements. This merges definitions, diagrams, and the structure of ordered sets.
 

Final Summary (In Long Form)

The Maths-III exam paper is carefully structured to test a wide range of mathematical knowledge expected from a 4th-semester B.Tech student. Section–A checks whether you have a solid foundation of transforms, discrete structures, logic, and algebra. Section–B tests your ability to write long, well-reasoned answers involving derivations, proofs, and transformations. Section–C demands higher-order thinking by combining transforms, differential equations, logic, induction, probability, and set theory. A student who prepares each area with conceptual clarity, practice, and writing discipline will perform well in this paper.