(SEM I) THEORY EXAMINATION 2019-20 MATHEMATICS-I

This is the complete AKTU B.Tech 1st Semester Engineering Mathematics–I (KAS103) question paper designed as per the official 100-mark university exam pattern. It includes Section A (short questions), Section B (medium questions), and Section C (long questions), covering all major topics of differential calculus, matrices, vector calculus, multiple integrals, differential equations, and coordinate geometry.

The paper tests a student’s conceptual understanding, problem-solving ability, and application of mathematical tools used throughout engineering. It contains a mix of theoretical proofs, calculations, transformations, vector identities, gradient, divergence, Jacobians, curve tracing, and calculus-based results.

This question paper is extremely useful for exam revision, practice, internal assessment preparation, and understanding the exam format followed by AKTU.

Section-wise Breakdown

SECTION A – Short Conceptual Questions (10 × 2 = 20 Marks)

This section includes quick, exam-focused questions that test fundamentals:

Linear independence of vectors

Lagrange’s Mean Value Theorem

Jacobians and transformation of variables

Divergence / solenoidal vector fields

Rank of a matrix

Double integral evaluations

Gradient at a point

Trigonometric inverse functions

Area bounded by curves

Parametric differentiation

Perfect for rapid revision before the exam.

SECTION B – Medium-Length Questions (Attempt Any 3 × 10 = 30 Marks)

This part includes proofs, theorem verification, transformations, and vector calculus:

Verify Cayley–Hamilton Theorem and find inverse of a matrix

Differential equation proof for y=emcos⁡−1xy = e^{m \cos^{-1} x}y=emcos−1x

Relation involving three variables using partial derivatives

Changing order of integration to simplify double integrals

Verification of Stokes’ Theorem for a rectangle in xy-plane

These questions strengthen theoretical concepts and derivation-based understanding.

SECTION C – Long Answer, High-Weightage Questions (5 × 10 = 50 Marks)

Each question has two alternatives:

Q3 – Matrices / Linear Equations

Conditions for unique, no, and infinite solutions using λ and μ

Finding rank of a matrix using row/column transformations

Q4 – Differential Calculus

Verify Cauchy’s Mean Value Theorem for exe^xex and e−xe^{-x}e−x

Curve tracing for polar curve r2=a2cos⁡2θr^2 = a^2 \cos 2\thetar2=a2cos2θ

Q5 – Advanced Partial Derivatives / Optimization

Relation between partial derivatives for a multivariable function

Largest rectangular box inside an ellipsoid

Q6 – Double & Triple Integrals

Double integral over a parallelogram using transformation u=x+y,v=x−2yu = x + y, v = x - 2yu=x+y,v=x−2y

Volume bounded by curves y=x2,x=y2y = x^2, x = y^2y=x2,x=y2 and planes z=0,z=3z = 0, z = 3z=0,z=3

Q7 – Vector Calculus

Verification of Divergence Theorem

Directional derivative & greatest rate of increase of a scalar field

These questions prepare students for long, step-by-step mathematical solutions required for scoring full marks.