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.
