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∬SF⋅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.
