(SEM I) THEORY EXAMINATION 2022-23 ENGINEERING MATHEMATICS I

This question paper evaluates the student’s conceptual knowledge, analytical skills, and mathematical techniques essential for engineering. The paper allows answers in Hindi, English, or a mixed bilingual format, giving flexibility to students.

The exam consists of three structured sections—Section A (Very Short), Section B (Short/Medium), and Section C (Long/Analytical). The questions assess understanding in Matrices, Vector Calculus, Multivariable Calculus, Differential Identities, Partial Derivatives, Jacobians, Double/Triple Integrals, Curve Tracing, Error Analysis, and Theorems such as Gauss Divergence, Stokes, and Cayley–Hamilton.

SECTION A — Very Short Answer (2 × 7 = 14 Marks)

Seven compulsory questions test fundamental concepts and stepwise logic:

Hermitian & Skew-Hermitian Matrices

Students prove that if A is Hermitian, then iA becomes Skew-Hermitian.

Eigenvalue Calculation

Given a matrix and an eigenvector, find the corresponding eigenvalue.

Partial Derivatives & Identities

Verification of identities involving

inverse trigonometric functions

composite functions

multivariable trigonometric expressions

Error Analysis

Percentage error in the volume of a rectangular box when each side has measurement error of 1%.

Double Integration

Evaluating ∫∫y dx dy over the region bounded by a line and a parabola.

Vector Calculus

Finding curl of a given vector field.

This section tests students’ clarity in basic definitions, formulas, and small computations.

SECTION B — Short/Medium Descriptive (7 × any 3 = 21 Marks)

Students must attempt three questions from five. These require multi-step derivations, algebraic manipulation, numerical skills, and calculus-based proofs.

Topics include:

Cayley–Hamilton Theorem

Verification for a 3×3 matrix and using the characteristic equation to find the inverse.

Higher-order Derivatives

Proving nth-order relations for variables defined implicitly with logs and exponentials.

Taylor Series in Two Variables

Expanding a multivariable function around (1,1) up to the second degree and applying it for numerical approximation.

Changing Order of Integration

Evaluating a definite double integral by converting limits.

Directional Derivatives

Using gradient vectors to find the directional derivative along outward normal to a sphere.

This section tests application, manipulation, and interpretation skills.

SECTION C — Long/Analytical Questions (7 × 3 = 21 Marks)

Students answer ONE question from each group. These problems involve deeper understanding and more advanced methods.

GROUP 1

(a) Consistency and Solution of Linear System

Using matrix method and rank analysis to test consistency and solve 3-variable simultaneous equations.

(b) Eigenvalues and Eigenvectors

Finding all eigenvalues and corresponding eigenvectors for a 3×3 matrix.

GROUP 2

(a) Curve Tracing

Tracing the curve

x2y2=(a2+y2)(a2−y2)x^2y^2 = (a^2 + y^2)(a^2 - y^2)x2y2=(a2+y2)(a2−y2)

in the xy-plane using symmetry, asymptotes, regions, and intercepts.

(b) Multivariable Differential Identity

Proving an identity involving second-order partial derivatives of an implicitly defined trigonometric function.

GROUP 3

(a) Jacobian and Functional Dependence

Finding the Jacobian of three given functions and proving they are not independent, then finding the relation between them.

(b) Optimization using Lagrange Multipliers

Finding dimensions of an open-top rectangular box of fixed volume that requires least material.

GROUP 4

(a) Triple Integration

Evaluating

∭R(x−2y+z) dz dy dx\iiint_R (x - 2y + z)\, dz\,dy\,dx∭R​(x−2y+z)dzdydx

over a region defined by multiple inequalities.

(b) Dirichlet’s Integral

Using Dirichlet’s formula to evaluate

∭xyz dx dy dz\iiint xyz \, dx\,dy\,dz∭xyzdxdydz

over the tetrahedral region bounded by coordinate planes and x+y+z=1.

GROUP 5

(a) Gauss Divergence Theorem

Applying divergence theorem to compute

∬SF⋅n^ ds\iint_S \mathbf{F}\cdot \hat{n}\, ds∬S​F⋅n^ds

for a vector field over a cylindrical surface.

(b) Stokes’ Theorem

Using Stokes’ theorem to compute the line integral of a vector field over the boundary of a triangular plane region.