(SEM-I) THEORY EXAMINATION 2018-19 MATHEMATICS-I

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.