This is the complete AKTU B.Tech 1st Semester Engineering Mathematics–I (KAS103) question paper designed as per the latest university examination pattern. It consists of three well-structured sections (A, B, and C) that test a student’s conceptual understanding, problem-solving skills, theorem application, and analytical thinking across core mathematical topics.
The paper covers Matrices, Differential Calculus, Vector Calculus, Multiple Integrals, and Differential Equations, making it extremely valuable for exam practice, assignments, and revision.
SECTION A – Short Answer Questions (10 × 2 = 20 Marks)
This section checks basic understanding and formula-based concepts:
Rank of matrices
Eigenvalues & matrix powers
Stationary points of multivariable functions
Cylindrical coordinate transformations
Definition of gradient and ∇ operator
Calculation of gradient at a point
Line and surface integrals
Rolle’s theorem
Partial derivatives of implicit functions
Taylor’s theorem for two variables
Percentage error/approximation problems
These questions strengthen conceptual clarity and build foundation for advanced problems.
SECTION B – Medium-Length Questions (Attempt Any 3 × 10 = 30 Marks)
This section contains theorem-based problems, matrix algebra, transformations, and verification questions:
Cayley–Hamilton Theorem
Finding inverse of a given 3×3 matrix
Expressing a large polynomial B(A) as a quadratic in A
Legendre-type Differential Equation
Proving the recurrence relation for sin(m sin⁻¹ x)
Evaluating yₙ at x = 0
Symmetric Functions of Roots
Roots of cubic equation involving parameters
Evaluate e(u, v, w)
Double Integrals → Polar Coordinates
Solving ∫∫ e^(−(x²+y²)) dx dy using substitution
Showing ∫₀^∞ e^(−r²) dr = √π / 2
Divergence Theorem
Verification over a cube for vector field
F = (x – yz)i + (y² – zx)j + (z² – xy)k
These questions involve rigorous mathematical techniques expected in university exams.
SECTION C – Long Questions (Attempt Any 1 from Each Set) (5 × 10 = 50 Marks)
Q3 – Matrices
Find inverse using elementary row transformations
OR
Reduce matrix to normal form & find rank
Q4 – Theorems of Calculus
Solve differential equation involving sin⁻¹ functions
OR
Verify Lagrange’s Mean Value Theorem for f(x) = x ln x on (2, e²)
Q5 – Maxima/Minima & Partial Derivatives
Minimum distance from point to a sphere
OR
Prove identity involving composite trigonometric function u
Q6 – Double Integrals & Geometry
Change order of integration & evaluate
OR
Volume bounded by planes x=0, y=0, x+y+z=1, z=0
Q7 – Vector Calculus
Show given vector field is both solenoidal & irrotational
OR
Find directional derivative of
Ø = 5x²y – 5y²z + 2 – z²x
at P(1,1,1) in direction of a given line
These questions represent the high-weightage topics that frequently appear in AKTU semester exams.
