(SEM III) THEORY EXAMINATION 2021-22 MATHEMATICS-III

This is the B.Tech Semester III examination paper for MATHEMATICS-III (KAS303).
The exam carries 100 marks and covers key mathematical tools required in engineering, including Laplace transforms, Fourier transforms, Z-transforms, discrete mathematics, logic, group theory, Boolean algebra, and mathematical induction.

The paper is structured into three sections — A, B, and C, progressing from short concept questions to analytical and long-form reasoning problems.

SECTION A – Short Answer Questions (20 Marks)

This section contains 10 brief questions (2 marks each), covering:

Laplace transform of an integral

Inverse Laplace transform of an exponential ratio

Modulation theorem

Z-transform of f(k)=1kf(k)=\frac{1}{k}f(k)=k1​

Logic problem: showing a statement is a contingency

Proving every subgroup of an abelian group is normal

Counting bit strings of length 8

Inverse function of f(x)=3x−2f(x)=3x-2f(x)=3x−2

Maximal and minimal elements of a poset

Boundedness (NULL) law: a∗0=0a * 0 = 0a∗0=0

This section checks fundamental theory, definitions, and basic problem solving.

SECTION B – Analytical / Medium-Length Questions (30 Marks)

Students must attempt any three questions (10 marks each). These include:

Solving simultaneous differential equations using Laplace transforms

Fourier transform of a piecewise function; Parseval’s Identity applications

Checking whether a ring defined by a∗a=aa * a = aa∗a=a is commutative

Solving difference equation yk−yk−1−6yk−2=−30y_k - y_{k-1} - 6y_{k-2} = -30yk​−yk−1​−6yk−2​=−30

Boolean algebra proof: equality using complements and distributive properties

This section evaluates engineering-level mathematical analysis and transformation techniques.

SECTION C – Long Answer / Advanced Questions (50 Marks)

Each main question has two choices, and the student must attempt one part from each (Q3 to Q7).

Q3 – Laplace Transform / Graphing

Graph and find the Laplace transform of a triangular periodic wave
OR

Inverse Laplace transform involving 1(p2+1)3\frac{1}{(p^2+1)^3}(p2+1)31​

Q4 – Heat Equation / Z-Transform

Temperature distribution in a semi-infinite medium (classic heat conduction problem)
OR

Solving second-order linear difference equation using Z-transform

Q5 – Logic & Arguments

Test validity of real-life logical argument involving graduation and going to Canada
OR

Translate statements into symbolic logic & test validity

Q6 – Number Theory / Induction

Counting integers not divisible by 5, 7, or 9 in the first 1000 numbers
OR

Proving a summation identity using mathematical induction

Q7 – Lattice Theory / Posets

Proving basic lattice identities (idempotent, commutative, associative, absorption)
OR

Showing that order-preserving functions are injective and constructing inverse order isomorphisms

This section tests advanced mathematical techniques, proofs, logic, set theory, discrete structures, and heat transfer mathematics.

OVERALL SUMMARY

The MATHEMATICS-III (KAS303) question paper provides a deep assessment of:

Laplace, Fourier, and Z-transform techniques

Differential & difference equations

Boolean algebra & logic reasoning

Ring theory & group theory

Discrete mathematics: sets, posets, lattices

Mathematical induction

Heat equation solutions

It blends theoretical concepts, derivations, proofs, numerical calculations, and logical reasoning, ensuring that students gain the mathematical foundation needed for advanced engineering courses.