About this previous year paper
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=a2cos2θr^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.